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Greetings!
I’ll spare you the lengthy introduction as you can just turn to, "Ask Mission #11 (2023 Greatest Hits #1 of #2)" to find that. You guys know the drill, so let’s get straight to the questions!
THREE-CARD POKER DEALER ERROR (PLAYER ADVANTAGE)?
Our first question involves an old-school Table Game, Three-Card Poker, which has been out for nearly my entire life. One would probably assume that a game that has been available for such a long-time, especially one that isn’t terribly complicated to begin with, would never have the dealers committing serious procedural errors.
But, as we all know, he who assumes misses opportunities to make money.
As you can see in this thread, Awz spotted a dealer error that probably had most WoV readers wanting to know where the casino is located:
Awz: While at a local casino I noticed the dealer at three card poker would consistently make the same payout error. Whenever the player folded and the dealer didn't qualify, instead of the player losing the ante, he would pay the player 1 to 1 on it. I'm curious as to whether or not someone would know what the new edge is with this change and if it's enough to flip the advantage. It's regular three card, ante bonus's are 1 to 1 for straight, 3 to 1 for trips, and 4 to 1 for str flush.
Mission: This reflects a tremendous advantage.
If we look at the WizardofOdds Three-Card Poker page:
What we immediately find is that the player will fold 32.5792% of hands, pursuant to optimal strategy, thereby incurring an expected loss of .325792 units.
I suspect that there would be mathematically optimal strategy changes such that players should actually fold more often. However, I am going to ignore that possibility as I can't think of an easy way to do it. Instead, what we will do is determine how often the dealer does not qualify.
Fortunately, this topic has already been addressed on these very forums, in which Tringlomane found:
Total qualifying combos: 15,380
Total combos: 52*51*50/6 = 22,100
Dealer qualifies: 15,380/22,100 = 0.6959 = 69.59%
Ergo, the dealer does not qualify 30.41% of the time.
You will notice that this is very close to the probability of the player folding. The reason for that is because the dealer qualifies with a Queen, but in certain cases, a player with a queen should fold.
The combined probability of the player folding (based on optimal strategy) and the dealer not qualifying is ROUGHLY:
.3041 * .325792 = 0.0990733472
However, this would be an expected swing of double that number of units because, not only are you not losing the one unit when you fold, but you are also getting paid an additional unit. Thus:
0.0990733472 * 2 = 0.1981466944
Thus, it is swing of roughly .198 units for every hand played.
The WoO page suggests that the normal expected return of the ante is -0.033730, so you end up with -0.033730 + 0.1981466944 = 0.1644166944
Which reflects an overall player advantage of greater than 16%. This return Table also takes play into account.
NOTES AND CLARIFICATIONS
1.) As I mentioned, this is all pursuant to the normal strategy for playing the game. However, that normal strategy would have you losing one unit, when you fold, 100% of the time, as is supposed to happen.
In this case, if you fold and the dealer does not qualify, then you will have not only NOT LOST one unit, you also get one unit in profits.
For that reason, a modified strategy would have you fold more often.
Based on the WoO page, we find that a player loses with a Flush (or less) 0.223805 or 22.3805% of the time. The player wins with a flush (or less) 0.213906 or 21.3906% of the time.
Of course, this wouldn't change making the Play bet and losing into a win, but what it does do is make some folds win more often where they wouldn't win at all normally. At a guess, you would fold a high-card queen (with no pair or better) more often than not. The reason why is because if the dealer qualifies and beats you (which he will more often than not) you would lose two units. However, by folding, you lose one unit roughly 70% of the time and profit a unit roughly 30% of the time whereas you should lose one unit 100% of the time. The normal strategy exists the way it does because folding is typically an automatic loss of a unit; that is no longer the case. In fact, folding goes from a guaranteed -1 to an expectation of -.4. You might actually fold some high card kings with no pair, or better, I don't know.
2.) Technically, you not having a playable hand normally makes it slightly more likely that the dealer qualifies. The reason why is because, absent very bad queens, you've removed three cards from the deck that would normally help in the dealer not qualifying. That means there are fewer low cards for the dealer to get. If you fold very bad queens, then you have only removed two cards that would cause the dealer not to qualify and one that would cause the dealer to qualify.
That's another reason that you would fold more queens with no pair or better, if not all of them. I could probably figure it out but it would take a long time.
Of course, we’re all human and we all make mistakes. My favorite human mistakes are human casino mistakes that make human advantage players money.
As you can see from my response, there are probably adjustments to be made that would get you to an even greater advantage than the 16.44% PLAYER ADVANTAGE I was able to determine following the Basic Strategy for the game, but an opportunity such as this isn’t likely to last very long, so you’d want to just take your 16.44% edge and play out that dealer’s shift.
After that, you can go home and try to figure out something more optimal; although, I would hasten to add, making plays that would look strange (when you had been playing basic strategy) might actually serve to alert the pit to the error the dealer is making faster.
It’s probably best to just take the 16.44% edge and be happy with that. It won’t take too long for the error to get corrected, but on such a low variance game, your advantage wouldn’t take long to be insurmountable for the house and result in great profits.
HOW AGGRESSIVE ARE CASINOS WITH TRESPASSING ADVANTAGE PLAYERS?
This is more of a personal experience question than it is a math question, but perhaps those of you who go to land-based casinos in the United States might find it useful:
McSweeney: I've heard hearsay stories of advantage slot players being kicked out of casinos, but I have to wonder if something else is going on there. I wonder if those people are particularly obnoxious about it, like sitting at a machine all day long staring at a bank of machines behind them waiting for people to get off (and thereby taking up the seat of the slot machine they aren't even playing) and possibly getting aggressive with the people playing by asking them to get off, or being aggressive with other people who are also camping and risk taking "their" machine once the ploppy quits. I find it hard to believe that a casino would just kick out a normal person who checks machines for good plays in a non-intrusive way. I mean sheesh slot manufacturers are making machines these days that literally display what the current collection number is for each bet on the bet screen to entice people to play it! How can a casino kick out someone for doing what the slot machines are designed to do?
Mission: I've been kicked out of several casinos and I'm not as active in this realm as most other people are. Of course, I knew I was coming in with some potentially hot cards on two different occasions. On one of those occasions, it was really more a question of when than it was a question of if.
The funny thing about that play was someone came out to a non-LV casino all the way from LV (after everyone else on the play had gotten popped already---except for the people who had to leave to do other things anyway) and he got popped on his first or second card. Can you imagine that? That's fine. I got popped on two cards in Atlantic City once---and one of them was mine!!! It's all in good fun. I like driving and blasting music anyway.
Of course, if you're talking about vulturing, I have never been kicked out ONLY for vulturing. There have been credible stories about people getting kicked out for vulturing at a few of the major corporation properties in Las Vegas, most frequently in high-limit rooms. I will say that if people are doing the sort of things in the examples of questionable acts that you offer, then they didn't get kicked out for vulturing, they got kicked out for vulturing AND being buttholes.
In my opinion, doing things with an eye towards running people off of machines is straight up dirty pool; that's to say nothing for actual creators. If there is action taken against vultures, just on general principle, then I would say people like that are the reason why such actions are taken. In my opinion, it's just bad policy to make the recreational players at all uncomfortable, hover over them, etc... If the Regal pops for them, then it pops for them. Good for them. I hope it's a top-tier result and that they don't end up giving it all back.
I've known a couple of people (though rare) that actively get angry when recreational players hit decent wins. Why? If we're advantage players for a living, or for part of our living, then neither the recreational players nor casinos owe anything to us. I suppose that casinos should, in theory, owe it to us to deal with us fairly as long as we are playing by the rules, but that's about it.
Basically, my experience is, when you’re engaged in multi-carding (taking advantage of profitable players club programs to an even greater extent by getting multiple cards—with permission of the people—who you usually pay—of course), then it’s a matter of when are you getting banned, not if.
When it comes to just taking advantage of variable-state slot machines, this is going to vary a ton by casino (with the Las Vegas Strip being most notorious, especially high-end properties), but generally, you’d only ever catch any grief in high-limit rooms. I think I’ve only been hassled by casinos once for just vulturing. Believe it or not, I’ve been trespassed from more stores (like, literal convenience stores) for taking advantage of Pennsylvania Skill Games than I have casinos.
One other variable that you probably wouldn’t have any way of knowing is if there’s anything else going on at that casino. For example, if they’re getting hit hard by multi-carders, then it’s possible that a casino will give you the boot thinking you are working with those guys. If there are a lot of creators, (people who claim to be helping, but are actually working to get people to leave advantageous plays on a machine for their confederates to take) then a casino might think that you’re wrapped up in all of that.
Generally speaking, if you’re not violating anyone’s personal space, or otherwise making a nuisance of yourself (and I know the good readers here would never think of that), then you should be fine, except for in high-limit rooms on The Strip, but how often are most people there, anyway?
ADVANTAGEOUS PAYTABLE ERROR
We’re just loaded with casino errors today! If you find a Video Poker paytable error that presents an advantage, what’s the best way to hit it?
TableforOne: Let's say you find a +EV machine at a small local casino. Multiplier promos make it worth as much as $120/hr to play at certain times. You are a regular at the casino and they know you well as a savvy gambler and high-volume VP player.
What would you do to get as much as possible from the machine to minimize risk of the casino reducing the paytable on this machine?
Thanks for any advice!
Mission: I wouldn't be shocked if the casino staff barely knows what the payback is or would ever bother to change it, but I could be wrong.
Once upon a time, after accounting for backend and other promos (drawing, earn x points games), with the multiplier days, I could play a 98.44%, 98.45% or 98.91% VP game, but with what I was getting back, the effective player edge ended up being 3-5% at any given time. The 98.91% became 100.91% on the points multiplier alone. The multiplier days actually changed (fewer days, weirder hours, lower multipliers---in that order) three different times before they actually changed the paytable on the machine.
Even without the multiplier days, back end still gave you no less than a 1% edge on a card you were using on an as-going basis...which is assigning a value of $0.00 to the room and food comps you'd get (which is the value I assign to those) and a much higher edge considering the backend on a clean card and just running off the free play until gone.
The multiplier days also applied to Bubble Craps for a bit, but it didn't take them too long to change that so it didn't earn points.
Still, those VP paytables stayed the same for probably ten years. It wasn't until post-COVID that they reduced them, I think, but it might have been before that.
ADDED: One nice thing was these units were multi-game, so also had Keno...which is what most people played. That probably disguised what I was doing a bit, but I agree with everyone else that you should just pound them while they're there to pound.
I’ll expand on that a little bit.
Basically, if the paytable is +EV by itself, what I would do is hit it hard and fast. I don’t think there’s much of a reason to be ultra-conservative about how hard you’re hitting it, because you might go there one day just for that paytable to be gone.
However, if you’re working alone, and there are multiple units with that paytable, what I would recommend doing is spreading your play out to different machines. The reason that I would do that is because, if the paytable is advantageous on its own, and you’re playing the same machine every day, then it’s more likely that the machine’s internal report will show the machine taking a loss, that the casino will figure out why and will also notice that you’ve been hammering that particular unit. In theory, that could lead to them trespassing you.
On the other hand, if the casino has five different units with that paytable, then by spreading your play out, you hopefully create the opportunity for other players to play different games that are not advantageous, or alternatively, to play the advantageous game poorly such that they would lose money in the long run.
When this happens, the internal report is more likely to show that the machines are profitable, so it’s a bit less likely that the casino will be alerted to the problem as quickly. Moreover, since it won’t look like you’re sitting there just beating the hell out of one machine all the time, even if the casino does notice the paytable error, they might not necessarily catch on to how much you were playing that unit and that game.
That’s the reason that I mentioned the multi-game units (with Keno games). I suspect one of the reasons my plays at that casino lasted so long is that, since most people played Keno on those units (8% average House Edge on those specific ones), the casino never really noticed that they had a high-paying Video Poker game on there that could easily be combined with multiplier days and mail. Eventually, they figured it out, and now the paytable is some 96-something percent garbage.
DO SPORTS BETTING SITES CHANGE ODDS FOR PROMOTIONS?
The short answer to this question is: Absolutely YES, sometimes.
One casino notorious for doing this, that’s actually regulated in some U.S. states, is DraftKings. As you will see in this exchange with OdiousGambit, DraftKings was offering a same-game parlay boost, which is great, except the base odds were much worse than at their competitors.
OdiousGambit: (100*100/155 + 100) * 100/211 + (100*100/155 + 100) = 242.49
if that means bet $100 , win $142.49, then that is +142 right?
that would mean DK is increasing the house edge
Mission146: They probably are. They give Odds Boost promotions on parlays all the time, so there probably is more base juice on parlays there as a rule.
Just do a comparison and find out. Let's go to two different sites and set up a no-promotion parlay at each. We just need to find something where the base lines are the same (or close) and then we will know. I guess we won't know as an absolute rule, but we will know it sometimes happens, which I'd assume anyway.
On PlaySugarhouse, I can take Falcons +3 at -110 and same-game parlay it with Christian Kirk to score an anytime TD at +160 to get +600.
On DraftKings, I can take Falcons +3 at -108 and same-game parlay it with Christian Kirk to score an anytime TD at +170 to get +500.
Wait a minute. What? If I am getting a better price on both the Falcons (-108 is better than -110) and on Christian Kirk (+170 is better than +160) individually,, then why is Sugarhouse going to pay me +600 while DraftKings is only going to pay +500 on a SGP? That's a difference of $100 on a $100 bet. Pretty significant.
DraftKings is probably offering a profit boost on SGP, or maybe a profit boost in general. When they do, it's usually 25%, so that effectively would make the line +625.
Ergo, the line would be better than what Sugarhouse is offering, but that makes sense, because both of the lines are individually better for the player (as if single bets) than the ones at Sugarhouse, so logic would imply that the parlay should pay more than the one at Sugarhouse.
That's why it's important to always shop and to give these things consideration even when you are trying to bet a promotion straight up. Considering that DraftKings is effectively offering me +625 on this (considering the boost), but Sugarhouse is going to give me +600 without a boost, that tells me that the DK boost probably isn't actually giving me positive value on this particular bet.
One thing to watch out for at ANY online casino, but especially sports betting promotions, is that they are really good at offering promotions that create the perception of value, despite the fact that the Odds likely remain in their favor.
That’s also why, if you have any questions about specific casino promotions, then you should come ask me on WoV. Depending on our relationship with the casino, I might even write an entire article analyzing a promotion for you and post it over here.
Given that this is a state-regulated online casino, OdiousGambit would later wonder if this is legal—which it absolutely is! Honestly, other than being slightly more reliable not to hold your withdrawal requests for a week or two, state-regulated casinos are little different from offshore ones, in my opinion.
OdiousGambit: Mission, gotta have time to follow up. And get my picks in too! You don't have to send a reminder this week, I'm on it
I will say that betMGM was not increasing the house edge on parlays when I tested them maybe 2 yrs ago. Is a site obligated to keep it honest? I don't know, but transparency would be appreciated!
Yes I started checking DK for a reason*, but my recent bad results are due to simply losing a bunch of bets ... better payoffs wouldn't have helped! but I am not going to continue with parlays if they skim off even more edge [unless the offer is for SGPs as a lot are]
Mission: They're not, 'Being dishonest,' is the thing.
The lines can be whatever they want them to be. The parlay lines can be whatever they want them to be. If they want to parlay a -108 and a +170 and give you an SGP line of +220, then they haven't lied, they've just given you an obviously ridiculous line. They don't have any sort of legal requirement NOT to give clearly ridiculous lines, but fortunately, they do (generally) have other books to compete with. Even where they don't; I don't think they would make the lines that MUCH worse in New Hampshire, where they literally have ZERO in-state competition, than elsewhere. They're probably the same lines.
DK has exclusivity there because they give the state either 49% or 51%; I forget which of the two gets the 51%. That's significantly higher than the online sportsbook revenue tax anywhere else.
Anyway, it's just for a person to know that promo exists and use it if they want that particular parlay. It's also for them to comparison shop. If I was goofing off with those things, then what I would do is assume the handicappers know their stuff and would create one where I have it significantly better than what the other books are offering without promotion. +625 v. +600...isn't that.
In general terms, it’s always good to compare promotions when you can to make sure that you’re getting the best possible deal. That’s especially true when it comes to sports betting, because it all depends on the odds that a sportsbook is giving you. Even when they’re offering a promotion, on rare occasions, the straight up odds (without taking a promotion) will be better at a different book than the odds you end up with when taking a promotion.
HOW ARE CRAPS ODDS CALCULATED
What always baffles me is some people, through no fault of their own, look at the math behind Craps and Roulette as some Universal Mystery to be solved.
Honestly, nothing could be further from the truth. Blackjack, for example, is a game that has WAY more variables (because there are several different combinations of rules), and would still be much more difficult to analyze even if that weren’t true. Craps and Roulette never change, sometimes Roulette gets a third zero on the Las Vegas Strip, but it doesn’t change outside of that. In any event, here is how you could determine the Pass Line ways to win, the long way:
Bossdog: I can’t figure out how to calculate the true odds of winning on a pass or come bet, ie bet $25 on pass. I know I house edge is 251 to 244 so 1.41% but how is that calculated. Obviously it includes winning ob 7/11 and losing on craps before a point. Help.
Mission: Are we doing homework? JK. It's fine; I wrote research papers in college for cash, anyway.
The first thing that we note is that we immediately win if the Come Out roll is either 7 or 11. Your example says bet $25, so we might as well.
(2/36 + 6/36) * 25 = 5.55555555556
The second thing we note is that we immediately lose if the Come Out roll is 2, 3 or 12:
-(1/36 + 1/36 + 2/36) * 25 = -2.77777777778
The remaining numbers are 4, 5, 6, 8, 9 and 10. If we roll a Point Number, then the only numbers a Pass Line bet cares about are that Point Number and 7. If you roll the Point Number again, baby gets a new pair of shoes; if you don't, baby is walking home barefoot.
ROLLS:
4 OR 10: 6/36
5 OR 9: 8/36
6 OR 8: 10/36
You can also sum these with the other Come Out probabilities from above to prove 36/36; I do not know how anal the teacher is. JK
At that point, fours and tens are 3/9 to win and 6/9 to lose. Remember, other results just mean we get to stand by a Craps Table longer. We should be very happy to stand there longer; you cannot see 18 Yo's in a row unless you stand by a Craps Table for a significant period of time.
(3/9 * 25) - (6/9 * 25) = -8.33333333333
Above reflects the expected loss from a Come Out Roll of either four or ten. What we have to do now is multiply that by the probability of such a CO. CO is also an abbreviation for Correctional Officer, but college students usually won't have to worry about those. At least...not for anything serious. We should all have a little fun from time to time.
-8.33333333333 * 6/36 = -1.38888888889
The next thing we have to look at is fives and nines. However, if you complete your Stats & Prob courses, as well as other courses, you might have a nice nine to five. That's one way to try to make a living. Dolly Parton has imparted this knowledge unto us; may we carry it with us and use the wisdom passed on by that large-breasted muse, for all time.
(4/10 * 25) - (6/10 * 25) = -5
-5 * 8/36 = -1.11111111111
Finally:
(5/11 * 25) - (6/11 * 25) = -2.27272727273
-2.27272727273 * 10/36 = -0.63131313131
We have already accounted for winning rolls during which we have established a point number. Our only remaining positive Expected Outcome is a Come Out winner, so we must subtract from its Expected Value the Expected Value of all of our barefoot outcomes.
5.55555555556 -2.77777777778 -1.38888888889 -1.11111111111 - 0.63131313131 = -0.35353535353
That means, for every $25 Pass Line bet, we expect to lose $0.353535353535353 cents. I hate change, so this is fine with me.
-0.35353535353/25 = -0.01414141414
Voila! This converts to a House Edge of 1.41414141414141414141414141414141414141414141%. Or, just 1.41%, if you prefer.
Hey, if you play Craps you might go home barefoot, but always give your dealers shoes on their toke bets.
At least it was fun to do.
If you’re interested in doing your own gambling math, and don’t know how to program, then I would suggest that figuring out how the Odds work for Craps and Roulette bets, every single one of them, would be a great place to start and get you acquainted with a few basic concepts.
CRAPS BETTING SYSTEM QUESTION
Let’s get something straight: I LOVE getting betting system questions. I especially love them when the system is not so complicated as to require a computer program to detail every possible outcome, so if you have a relatively uncomplicated betting system that relies on a binary outcome (win/loss, or close enough to), then by all means, come on over to WizardofVegas, create an account and ask it in a thread.
Why do I love these betting system questions? Because I already know the answer; the answer is whatever the House Edge of the bets are multiplied by your average bet size being your expected loss. That’s the only answer, but the fun is in getting there, as you will see:
Ciabelle: I stumbled across a dead simple betting strategy which I've found very little online about. It's a slight variation on the Doey-Don't (equal bets on the pass and don't pass lines) but, unlike that strategy, it can actually win money with very little risk.
Simply put, you bet the don't pass on the come out. If a point is established, you place an equal bet on the point. Adjusting slightly for the 6/8 or for buying the 4/10. You break even on the 7, and win a small profit if the point is hit.
I'm not sure I calculated the odds correctly, but I'm showing a house edge of just under 3.24% but could be much closer to the general EV for a DP line bet of 1.36%
Suppose we play the dark side for $100, we win on a 2 or 3, lose on 7, or 11, and push on 12. Easy peasy. But when a point is hit, that's where my math may be a bit off. I'm preparing for a cruise where the vig is collected up front on buy bets, so with this example, I'd buy the 4/10 for $105. I'd place the 6/8 for $102 to closely match the line bet. A matching bet of $100 makes sense for the 5/9. So in the following table I multiply the return if the point is hit by the number of combinations minus the loss from the line bet. Then do the reverse if the 7 hits (always 6 combinations) then divide the return by the total number of consequential outcomes for that point.
I believe these calculations themselves are accurate. I'm but not sure what unit to compare the return of ~$116.55 against the $3600 bet on the Don't Pass line for each of 36 possible rolls, or should I include the amounts wagered when a point is established? Which would make the house edge significantly lower.
In any case, this strategy is all about earning comps (e.g. free cruises) quickly over winning or losing much money. There's certainly a risk that the pit boss would notice your strategy and zero out your rating. It isn't my original strategy, but I found on an old youtube video by someone who claims to have been successful with it on a bubble craps machine.
Mission: What do you mean when you ask about, "House Edge?" Each individual bet has a House Edge, so I assume that what you want to know is the expected loss on your total action.
In order to determine that, the first (and easiest) thing we have to do is determine your average total bet. Using the bet amounts you have provided, here is our total bet for each outcome:
2, 3, 7, 11, 12: $100---> (100 * 12/36) = 33.3333333333
4, 10: $205---> (205 * 6/36) = 34.1666666667
5, 9: $200---> (200 * 8/36) = 44.4444444444
6, 8: $202---> (202 * 10/36) = 56.1111111111
Think of this like an expected bet amount, in total. Assuming we are going to behave this way every single Come Out, we can sum this up and determine our average expected total bet:
56.1111111111 + 44.4444444444 + 34.1666666667 + 33.3333333333 = 168.055555556
If we made it a binary by which we would bet $100 total 1/3rd of the time and $200 total 2/3rds of the time, then that would be an average of $166.67 (rounded) average bet, so this obviously checks out because of the little extra that is bet in a few instances.
EXPECTED VALUE OF INDIVIDUAL RESULTS
With that, we have to determine the EV of all of the individual possible results. After we have done that, we will be able to determine the total expected loss and compare it to our expected average bet amount.
The first thing that happens is that we lose $100 on rolls of 7/11, win $100 on rolls of 2/3 and Push on a CO of 12:
(1/36 * 0) + (100 * 3/36) - (100 * 8/36) = -13.8888888889
In other words, that is our Expected Loss as we isolate the Come Out roll.
Naturally, your betting system is very goal-oriented and its goal is to get a fair amount of action in whilst trying to reduce Variance. Otherwise, it would be an objectively terrible system because the entire thing forfeits the advantage on the DP bet you end up with on those occasions that the roll is a point number.
With that, let us look at the possible results for varying point numbers:
4 & 10
The first thing that we note is you establish a point of one or the other with a frequency of 6/36.
A result of seven results in a win for the DP, which is +$100, but then you lose the $100 from the 4/10 as well as the $5 commission because the commission is paid upon making the bet.
In the event that the 4/10 comes in, then you have lost $100 from the DP bet and have also lost the $5 for the commission, which I will treat as being part of the DP loss for simplicity. However, you have won $200 on the Buy bet, which essentially functions like an Odds Bet.
(3/9 * 95) - (6/9 * 5) = 28.3333333333
That is the expected result when we isolate that combination of bets, which we must now multiply by the probability of being able to make that combination of bets in the first place:
28.3333333333 * 6/36 = 4.72222222222
5 & 9
The first thing that we note is you establish a point of one or the other with a frequency of 8/36.
A result of seven is a win for the DP, which causes us to break even because we lose the $100 we have bet on the five or nine.
In the case of the 5/9 coming in, then you would get paid $140 for a profit of $40 because of the $100 DP loss.
(4/10 * 40) - (6/10 * 0) = 16
Once again, we have to have the opportunity to do this in the first place:
(8/36 * 16) = 3.55555555556
6 & 8
The first thing that we note is you establish a point of one or the other with a frequency of 10/36.
A result of seven is a win for the DP, which causes us to lose $2 because we are betting $102 on the Place 6 or 8.
A result of six or eight is a win of $119, but is a profit of $19 because we will have lost $100 on the DP.
(5/11 * 19) - (6/11 * 2) = 7.54545454545
7.54545454545 * 10/36 = 2.09595959596
THE RESULT
With that, we simply need to add all of our results with an expected profit to our expected loss on the CO roll:
-13.8888888889 + 4.72222222222 + 3.55555555556 + 2.09595959596 = -3.51515151516
That number reflects our expected loss on the overall proposition if we do it exactly as you suggest you will. We can compare this expected loss to our average expected bet and will arrive at an expected average House Edge for our total action:
3.51515151516/168.055555556 = 0.02091660405
In other words, you are expected to lose slightly more than 2% of all monies bet.
ANOTHER WAY
Another way we can confirm this is to simply multiply each bet amount, and the likelihood of our betting it, against the House Edge of each particular bet.
For example, we make the DP bet 100% of the time, so we would have:
100 * .0136 = 1.36
We are going to bet that 4/10, essentially for $105, 6/36 of the time. We also only care about bet resolved since we are playing these out to resolution:
105 * .0476 * 6/36 = 0.833
We are going to bet that 5/9 for $100 8/36 of the time.
100 * .04 * 8/36 = 0.88888888888
Finally, we are going to bet that 6/8 for $102 10/36 of the time.
102 * .0152 * 10/36 = 0.43066666666
Basically, this accounts for the expected loss of each of the individual bets and also factors in the probability that we make any of these bets in the first place, except for the Don't Pass bet, because we always make that. If we add these expected losses together, we get:
1.36 + .833 + .888888888888 + .430666666666 = 3.51255555555
This is slightly off from what we have above, but errors are due to rounding. It's within a fraction of a penny. If nothing else, what I did above is going to be substantially more accurate because, for this part, I was using House Edges that had already been rounded off.
The point is that the expected loss essentially matches also doing it this way and we would expect our average total amount bet to remain the same.
With that, I am confident to say that you expect to lose roughly 2.09166% of all monies bet.
In fact, do you know how much I enjoy questions like these? I enjoy them so much that, on occasion, I will actually suggest a better system. The system I proposed appears below; what makes it a better system is that the expected loss (relative to total action) is going to be lower because the way I would do what the OP wants to accomplish puts more money on Odds bets:
Mission: However, what if the table is 3/4/5x Odds? With that, I can make the $100 DP bet, $300 in Odds bets, and a $200 Buy 4/10 bet whilst paying $10 commission:
In other words, we have:
$100 DP
$300 Odds
$200 Buy
$10 Sunk Cost Buy Commission
If the number comes out a four, then I will lose $410, but I will make $400 on the Buy bet, therefore losing $10. If the number comes seven, then I lose the sunk costs of $210, but my DP makes $100 and I also make $150 on my DP Odds bet for a net profit of $40 on this.
(40 * 6/9) - (10 * 3/9) = 23.3333333333
23.3333333333 * 6/36 = 3.88888888888
We have a lower expected profit on this precise outcome, but we really have no reason to care about that if our goal is just to get action out there. For that reason, we can look at the total expected loss relative to our total amount bet each time we do this. We will ignore the initial DP bet for $100 because we make those anyway.
OP QUESTION:
105 * .0476 = 4.998
MISSION SUBMISSION
300 * 0.000 = 0
210 * .0476 = 9.996
Isolating just what we do after a four or ten is established, the OP's question suggests a bet of an additional $105 total with expected loss of $4.998. My alternative to this is to actually bet an extra $510, with $210 being the same bet that the OP is making, however the expected loss on my total action is twice what the OP's is. However, most of my action is on odds, so when we look at the average House Edge of my total action:
9.996/510 = 0.0196
In other words, the average House Edge relative to my total action is 1.96%, which is still bad compared to just making a DP bet, but that's because the Buy 4/10 with paying commission on bet is such a terrible bet in the first place.
Also, to be honest, I could see where you would fade the variance of sixes and eights (the House Edge of a Place Bet on these isn't that much worse and can probably be made the same---or less---for average House Edge relative to total action if made in conjunction with Odds bets on the DP) and because those are 5/11 to come as opposed to 6/11 for the DP to win. I just don't understand why we would want to make 4%+ HE bets (especially without making additional Odds bets to trim the average HE on the additional action down) when our DP bet is at a HUGE advantage in the first place.
Not to be rude, but I'd have to say that if the variance of wanting a seven vs. a 4, 5, 9 or 10 is too much to sweat, then this probably isn't a worthy pursuit in any case.
As everyone is certainly aware, Wizard always says, “All betting systems are equal-equally worthless.” From a House Edge standpoint, I agree with what Wizard says 100%. It’s undeniable. At the same time, players might occasionally have goals that are unrelated to the House Edge itself, and when it comes to accomplishing those goals, some systems may be more…let’s say, functional, than others.
As ever, if you’re a recreational player and are playing with money you can afford to lose, then just have fun! If you have a system question, come ask. As Wizard says, it doesn’t make a difference whether you use a system or not as the result will just be bet amount * House Edge. Still, I find them fun to discuss and if using a system will cause you to have more fun gambling, then go for it.
GETTING THE RANK YOU DISCARDED (VIDEO POKER)!
Few things are more annoying on Video Poker than discarding some cards and getting one of the same rank (such as a ten) back, or worse, two of the same rank!
While it might be annoying, is it uncommon? Is the game rigged? Let’s find out:
ChumpChange: Game King will always be sus to me. But I find the most obvious things to be I'll discard a 4 of something and get a 4 of something else back, that kind of thing. Is the RNG just that bit deep to only find the next suit down?
Mission: If you're holding two cards and discard one four:
nCr(3,1)*nCr(44,2)/nCr(47,3) = 0.1750231267345051
You will receive exactly one four 17.5% (approx) of the time.
And a pair of fours:
nCr(3,2)*nCr(44,1)/nCr(47,3) = 0.0081406105457909
About 0.814% of the time.
Between the two possibilities, that's more than 18.3%, or close to one in five hands, that you would expect that to happen. You could also get trip fours, but it's not very likely.
Naturally, the probability of seeing another four increases if you hold either zero or one cards.
As it turns out, what ChumpChange is suggesting should happen a lot, particularly if you play a good deal of Video Poker, which I believe he does.
When it comes to pretty much anything related to gaming machines (slots and Video poker) and, to a lesser extent, Table Games…it’s important to understand that our brains are hardwired to notice things that seem strange. That’s especially true if the thing in question happens multiple times within a short period.
Of course, that doesn’t mean that anything is rigged. When you’re looking at better than a 1 in 6 chance at a particular event occurring, given a particular situation, and you see that situation a fair number of times, then you are going to have periods where it seems like it’s happening a lot in short succession, because it should.
In fact, doesn’t it seem like, on Slots and Video Poker, when you’re almost out of money, you hit small wins that give you just a few more spins?
Risotto: I frequent the 7-11 which is a short walk from my home and has Game King machines that offer video poker options. My favorites are Duces Wild and Joker Poker. I have noticed what I believe are a few unusual patterns to the hands dealt. For example, I am hitting a pair of kings or better and three-of-a-kinds. Then lose a hand or two. Basically, keeping even at a steady pace. Then, if and when I hit a straight flush or a royal with a duce, this is followed by a run of very poor hands totally devoid of any decent cards to hold which promptly sucks back that decent win; and then some.
The second pattern while playing that I notice is that when you get close to running out of credits, the machine suddenly starts to spit out a bunch of smaller-sized winning hands before finally taking it all back. It's almost as though it has been somehow hardwired into the program to give you those smaller winning hands before losing it all so as to give you hope and encourage you to put more money in. Which of course they want you to do anyway.
I tend to play very fast. Nervous energy I guess. So, it is much easier to observe how dramatic these questionable fluctuations are. I have seen these patterns of play time and time again. Is it merely a coincidence? Is it really totally random even though it appears otherwise? Or, is there something sus going on here?
Mission: The second paragraph is confirmation bias. Most of the hands that you win are small-sized wins because those are the most probable hands that are simultaneously winning results. If you will kindly look here:
I just picked a paytable, so it's probably not the same, but let's add up the probabilities of getting any winning hand that pays as much, or less, than a 4OaK.
0.064938+0.021229+0.016784+0.056070+0.284690 = 0.443711
With that, you win roughly 44.3711% of hands with a four of a kind, or less. You lose roughly 54.6897% of hands. That leaves only 0.9392% of hands that can have any other result.
Given those probabilities, the most likely result with four, or so, total bets on the machine remaining is that you will win a few smaller hands before ultimately busting out. If you had only four more hands worth on the machine, you'd actually be less than 10% likely to lose all four of those hands without winning any.
As you can see, Risotto described a situation that is, in fact, significantly more likely than not to happen! Again, it seems unusual to us because we’re there in the moment, hitting all of these small hands when we either want to hit something decent, or perhaps just run out of money so we can go take the leak we’ve been holding for the last five minutes.
Our brains are so hard-wired to detect abnormalities and patterns that we sometimes fail, in the moment, to recognize when an event is actually extremely likely to occur!
WRITING OFF OPPORTUNITIES TOO EARLY!
If you’re ever going to get into slot advantage play at land-based casinos, the biggest piece of advice that I could give anyone is, “Identify as many opportunities as possible.”
Especially these days, competition for variable-state plays is pretty tough, sp the worst thing that you can do is limit your own number of opportunities by not trying to figure out plays, or in some cases, disregarding machines as unbeatable too early. DarkOz and I had the following discussion:
DarkOz: I ran across a new variant of Scarab. Unvulturable in my opinion.
So it's a mix of Scarab and lock it link.
Ten spin cycle with latent Scarabs landing.
Tenth spin the Scarabs at random switch to EITHER a $ amount(including mini, minor, major etc) with a blue background OR wild.
If six of these latent scarabs turn into blue background $ then the tenth spin enters lock it link mode and you play until three losing spins (no symbols land) You then win the lock it link final amount plus any money from the wild symbols in winning positions.
However the blue background lock it link symbols are blockers and don't pay unless they add up to six positions.
So let's say left to right is Wild, $, Wild, that equals a zero win because the wild symbols are separated by the lock it link.
The end result is every tenth spin you win back practically nothing unless you get the lock it link activated which playing for about twenty minutes happened once for me.
In addition on one spin I had about 8 or 9 Scarabs literally all lined up at the left and got zero as just enough turned into lock it links to block any win from the wilds but only 5 turned lock it link to avoid the lock it link bonus
What a great way to make a fun game a piece of garbage.
Mission: I wouldn't be so quick to declare a game, "Unvulturable."
The first reason is that you have to look at the most extreme possible case, right? With Scarab, the most extreme case is that you're going to have a full screen of symbols and the next spin is the tenth...you obviously take that.
As with normal Scarab, I would assume that the RTP is heavily centralized on both tenth spins and Free Games. No matter how you slice it, the game gets itself to 80%+ (most jurisdictions require this) somehow, and if a variable state is how it does it, then that can be good for vultures.
I haven't seen one of these yet, but if I did, I'd also be inclined to look at the rules screen and see if the would-be, 'Wild' spots can even become money symbols with five, or fewer, to-be 'Wild,' spots on the board. I could also watch Youtube videos to maybe determine that or see if any of the Youtubers take the time to show each screen of the rules.
So, I don't think this game is going to have the same number of, 'Home runs,' that are absolute locks like original Scarab does...but if you can figure out when you should play it and others are inclined to ignore it completely, then that might pay off for you.
It sort of reminds me of attitudes towards another game, though this game had MORTAL LOCKS like the original Scarab can sometimes be. I forget the name of this game, but it resembled a mechanical slot (in terms of how the cabinet looked) and every time you would hit a wild symbol, you'd get wilds there on the next spin, unless the wild symbol hit only on the bottom.
The way the machine worked, the reels were such that WILD appeared in stacks of three and NEVER anything except stacks of three. If you hit a WILD on top only on a particular reel, then that reel would have Wilds for the next spins as follows:
- Top/Middle
- Top/Mid/Bottom
- M/B
- B
Because the three wild stack just slid one down every spin.
Anyway, I think that might have been the first machine I've ever seen that required you to have money in to look at different game states. Between that and the game being considered a low-value play, virtually everyone ignored it. I suppose they also didn't want to take the time to insert a ticket and cash it out every check with...I remember there being four..such machines to check at that location.
Hey, it can be as low value as it wants to be. If nobody else wants four spins with wilds on reel 2, then I'm more than happy to take them. You honestly almost never finished down money on many plays, but where you'd really make bank is if the Wild Stacks would appear on other reels as you were playing off a wild stack that you found.
My usual rule of thumb for the game was taking any next spin WILD on the first three reels, ignoring Reels 4 & 5 completely. Maybe 4 + 5 would have been good, but I don't know, because I never took those without there being at least one in the first three.
It’s similar to one of my favorite sayings, “If you don’t ask, then the answer is always ‘no.’” In this case, if you write something off as never being playable without doing the proper amount of exploration and analysis, then it won’t ever be a play…at least not one that you’ll get.
VIDEO KENO—IS THIS NORMAL!?
Ah, one of my favorite -EV games, Video Keno.
That’s especially true when it comes to Land Casinos and very small bets, like $0.10/play. I have no idea why, but for most online Keno games (when you can even find them) the House Edge tends to be much higher than in land-based casinos, which is weird, because online RTP’s are better for almost every other game, or roughly the same, for Table Games.
In any event, someone came up with a way to play Multi-Card Keno that involved picking the same four numbers on all cards, but then picking a different fifth number on all of the cards. This, ‘system,’ even though it’s not really a system, showed tremendous success. Was something wrong with the machine? Is this mathematically viable?
As it turns out, the results were actually fairly normal:
BrotherAron: 1. Is any of this even right? Have I just been getting lucky?
- How do I calculate the RTP of this strategy?
- If you are familiar with the Gambler’s Bonus system available at Yorky’s and many local Vegas bars, how do I factor points earned and won into the RTP?
Mission: 1.) You have just been, 'Running well.'
2.) You can use the Wizard of Odds Keno Calculator:
Zero of Five: 0
One of Five: 0
Two of Five: 0
Three of Five: 3
Four of Five: 13
Five of Five: 838
(Per One Unit Bet)
Return to Player: 0.949452446287889
You are correct in stating that an RTP of basically 95% is extremely generous by Video Keno standards.
However, the return of each individual bet doesn't care what numbers you pick. You can pick the same five numbers twenty times, pick the same four numbers (with the fifth always being different) or pick the same three twenty times (and make the other two always different); it doesn't matter.
In this case, what is influencing your short-term results is the fact that all of your cards are highly correlated.
But, let me ask you this: How well would you be doing if, instead of having the fifth number always be different, the fifth number was always the same? Naturally, you would be paid 838-FOR-1 twenty times. Of course, you would also be betting twenty units.
Think of it this way: If you inserted twenty units worth and hit that on the first draw, then your bankroll would be 16,760 units, as you don't get the original bet back. In other words, you could completely whiff on the next 837 draws and would still have the units you originally inserted to the machine.
The Difference in Variance
Another thing to note is that your Keno game is simply a higher Variance game when it comes to the five-spot. Let's compare the two games using the WoO Calculator:
Four Spot Returns (For One):
Two of Four: 0.425270931600046
Three of Four: 0.216239456745786
Four of Four: 0.306339230389863
SUM OF VARIANCE: 31.667243286177953
The four spot also has a slightly lower RTP, but not so much lower as it would explain the difference in short-term results by itself.
Five Spot Returns (For One):
Three of Five: 0.251805156868448
Four of Five: 0.157200394542167
Five of Five: 0.540446894877275
SUM OF VARIANCE: 454.792058559047575
In simple terms, Variance is the degree to which a particular result varies from the average expected outcome.
The reason that the Five-Spot Variance is remarkably higher is twofold:
1.) Five of Five is significantly less likely than four of four.
2.) Five of Five, relative to probability, pays significantly more than Four of Four. This must be true because both events have a fixed probability, but hitting Five of Five represents a greater percentage of the game's return. In fact, Five of Five represents a greater return percentage than the most likely event on a Four of Four game, which is hitting two of four.
In short run terms, given that this is a negative expectation game, Variance is your friend. That's true of any negative expectation game if the goal is to be profitable, at some point.
For example, a Keno game with no variance where you Pick Four with a 95% RTP would look like this:
Zero of Four: PAYS .95
One of Four: PAYS .95
Two of Four: PAYS .95
Three of Four: PAYS .95
Four of Four: PAYS .95
The Variance is zero because the result is always the same. Regardless of what hits, the result does not deviate from the mean expectation whatsoever. You bet twenty and you lose one of twenty units, which is a 5% loss.
With that, let's jump over to the Beating Bonuses simulator and simulate 10,000 attempts of both games. We will start with a bankroll of 10,000 just so that busting out is impossible, unless there is zero return on every single draw, which won't happen. This simulator isn't perfect, but the percentage of those who finish these 10,000 trials, I am extremely confident, won't even be close comparing four spot with five spot.
ENTRY: Blank (We have to do this so we can put a custom paytable)
FOUR SPOT SIMULATION
The first thing that we are going to do is just combine the probabilities of Zero + 1 because they don't pay, so it's functionally the same thing.
0.308321425410033 + 0.432731825136888 = 0.74105325054
That is the probability of a Return of -1. Again, it doesn't matter if we hit one ball, or zero.
Two of Four: 0.212635465800023
This will have a return of 1, which you can think of as +1.
(This game is on a FOR ONE basis, so we actually want all results to pay stated -1)
Three of Four: 0.043247891349157
This will have a return of four, rather than five.
Four of Four: 0.003063392303899
This will have a return of 99, rather than 100.
The Simulator calculates a 5.22% House Edge; that is good because that is correct.
We insert the following:
DEPOSIT: 10,000
BONUS: 0
WAGERING: 10000
BET SIZE: 1
SIMULATIONS: 10,000
RESULTS:
Average Return: (523.13)---(Close to House Edge, so this is good)
Minimum Return: ($2,622)
Maximum Return: $1,992
Chance of Gain: 18%
Chance of Bust: 0% (By design)
Standard Deviation: 568.37
FIVE SPOT SIMULATION
You'll have to take my word that I haven't already done this, but I am confident that this is a principle that will hold. The Maximum Return from Five-Spot Play will be greater, the Standard Deviation will be greater AND a greater percentage of players will finish ahead. Let's watch the magic!
The first thing that we will do is, again, combine all loss probabilities because they are functionally the same.
0.227184208196866 + 0.405686086065833 + 0.270457390710555 = 0.90332768497
Naturally, those will have a return of -1. Of course, the first thing that we notice is that we are significantly more likely to lose on an individual trial in this game. The hit rate is much lower. The paytable, by design (which is what makes Keno a good game for players who understand it---you can get the experience you want out of it), will absolutely destroy you if you don't hit your five spots.
THREE OF FIVE: 0.083935052289483
Again, this functionally pays 2 for the purpose of the simulator.
FOUR OF FIVE: 0.012092338041705
This pays 12 for the purposes.
FIVE OF FIVE: 0.000644924695558
This pays 837.
The simulator gets a House Edge of 5.05%, which is good, because that's what it is. You always want to verify that because..sometimes it's just flat out wrong, but usually, you may have put something in incorrectly.
We insert the following:
DEPOSIT: 10,000
BONUS: 0
WAGERING: 10000
BET SIZE: 1
SIMULATIONS: 10,000
Results:
Average Return: ($494)
Minimum Return: ($6,143)
Maximum Return $8,299
Chance of Bust: 0% (By design)
Chance of Gain: 38.18%
Standard Deviation: 2,126.6
CONCLUSION
You got lucky. Strictly, I don't believe in luck. If enough people played the exact way you did, then some of those people WOULD experience results that are at least similar.
Of course, having a highly correlate (four of five of the picks are the same) multicard reduces your variance slightly; the way to get the most variance, naturally, would only to be betting one card...or to bet on twenty that are all the same.
As we see in our results, despite the sub-5% House Edge, our worst performer in the 5 of 5 simulation lost 61.43% of all monies bet. BRUTAL!!!. How brutal? Well, if they hadn't hit a single Five of Five, but ran exactly as expected on 4/5 and 3/5, they would have done better.
In the meantime, our Four-Spot players get a smoother ride, but fewer than half of them (compared to Five-Spot players) are still ahead after 10,000 spins.
Why? Vairance---which is our friend in a negative expectation game...unless you just want to play as long as possible and reduce your risk of ruin within x (where x is a relatively small number) of draws.
When we look at the Variance, what we notice for the Four-Spot is that the SD is equal to almost the average amount lost.
What are the implications of that?
Well, you look at the 68-95-99.7 rule, which says that 68% of players (in this example) will finish within one Standard Deviation. In the case of the Four-Spot, the majority of the players who finish within one SD to the right side of the mean still lost some amount of money.
In comparison, a tremendous percentage of the players to finish within 1SD to the right of the mean will be ahead after 10,000 attempts. The reason why is because 1SD is substantially larger than the mean average result is. That also explains why 38.18% of players finished ahead in that simulation.
When we look at the rule, we see that the distribution, roughly, would look like this:
34%---1SD Left
34%---1SD Right
13.5%---2SD Left
13.5%---2SD Right
2.5%---3SD+ Left
2.5%---3SD+ Right
With that, we can get rid of 16% of our winners as being outside of the first Standard Deviation. That still leaves us with 22.18% of players in this simulation who finished ahead. We know that approximately 34% of players will finish one SD to the right, therefore, just over 65% of our players who finished 1SD to the right also profited. I didn't calculate for the Four of Four, but the old eyeball test says that only 2% of players within 1SD made a profit after 10,000 attempts, as 16%/18% would have been 2SD+ to the right results.
Anyway, that is your success both explained and demystified. I'm extremely happy for you as I happen to enjoy Keno as a game, but this is absolutely expected to happen for some players and you did not break mathematical reality; you simply got the favorable end of that reality. May your streak continue should you choose to continue to play!
But…I really do like Keno, so I wasn’t quite finished:
Mission: BrotherAron,
You know something? It occurs to me that you would probably be interested in knowing the probabilities of hitting particular numbers of Five/Five given that your shared Four/Four (between all Five-Spot Cards) has been hit.
Well, guess what, brother? Christmas came early for you this year, and as a present, I am going to do just that. I am going to do all of these as my gift to you, in order to spread joy, mirth and merriment.
But, you know how the old saying goes:
Quote:
If you give a man a fish, he still needs a pan.
I think so. Right?
Anyway, not only will I give you these fish, but I'm also going to share my cooking equipment (that doesn't actually belong to me) with you and teach you how to use it.
Let's have some fun with probabilities!
The first thing that we are going to need is a sexy, sexy, scientific calculator that makes combinatorics easy...but still a bit time-consuming.
Ladies and gentlemen, she even looks good in an elf get up, let's meet our scientific calculator:
She'll even let you push her buttons; just be classy about it.
Here we go:
FOUR OUT OF FOUR IN THE FIRST PLACE
nCr(4,4)*nCr(76,16)/nCr(80,20) = 0.0030633923038986
This is actually going to be easy since we're only looking at probabilities.
1/0.0030633923038986 = 1 in 326.4355005159993898556
Basically, we are going to take the four out of four for granted from now on. Besides that, if you want to figure out the probability for a specific number of five out of fives, then all you need to do is multiply any of the decimal probabilities (below) from the decimal probability of four out of four (above).
Okay, so assuming we have hit the four numbers we need, this is what remains:
NUMBERS COVERED: 20
DRAWS REMAINING: 16
COVERED NUMBERS THAT DO NOT HELP: 56
Basically, all of our combinatorial questions for the calculator are going to take this form:
(Numbers Covered, Numbers that Hit)*(DO NOT HELP NUMBERS, DO NOT HELP #'s HIT)/(NumbersCovered+DONOTHELPNUMBERS, TOTAL DRAWN)
The only variable inputs are going to be the number of each type of numbers to hit or not hit. Let's start with 16 out of 16 and work our way down:
REMEMBER THAT ALL PROBABILITIES BELOW TAKE HITTING THE FIRST FOUR FOR GRANTED!!! IN ORDER TO GET A COMBINED PROBABILITY YOU MUST ALSO MULTIPLY THIS BY THE FOUR OUT OF FOUR PROBABILITY!!! Also, all are rounded off.
nCr(20,16)*nCr(56,0)/nCr(76,16) = 0.0000000000004474 or 1 in 2,235,136,343,316.94***
nCr(20,15)*nCr(56,1)/nCr(76,16) = 0.000000000080168 or 1 in 12,473,805,009.48
nCr(20,14)*nCr(56,2)/nCr(76,16) = 0.0000000055115483 or 1 in 181,437,219.74
nCr(20,13)*nCr(56,3)/nCr(76,16) = 0.0000001984157381 or 1 in 5,039,922.79
nCr(20,12)*nCr(56,4)/nCr(76,16) = 0.0000042721388599 or 1 in 234,074.79
nCr(20,11)*nCr(56,5)/nCr(76,16) = 0.000059240325524 or 1 in 16,880.39+++
nCr(20,10)*nCr(56,6)/nCr(76,16) = 0.0005538970436498 or 1 in 1805.39
nCr(20,9)*nCr(56,7)/nCr(76,16) = 0.0035967340496738 or 1 in 278.03
nCr(20,8)*nCr(56,8)/nCr(76,16) = 0.0165224970406891 1 in 60.52
nCr(20,7)*nCr(56,9)/nCr(76,16) = 0.0542276825950821 or 1 in 18.44
nCr(20,6)*nCr(56,10)/nCr(76,16) = 0.127435054098443 or 1 in 7.84
nCr(20,5)*nCr(56,11)/nCr(76,16) = 0.213164090491941 or 1 in 4.69
nCr(20,4)*nCr(56,12)/nCr(76,16) = 0.2498016685452434 or 1 in 4
nCr(20,3)*nCr(56,13)/nCr(76,16) = 0.1989370754025467 or 1 in 5.03
nCr(20,2)*nCr(56,14)/nCr(76,16) = 0.1018368362179704 or 1 in 9.82
nCr(20,1)*nCr(56,15)/nCr(76,16) = 0.0300150675168755 or 1 in 33.32
nCr(20,0)*nCr(56,16)/nCr(76,16) = 0.0038456805255997 1 in 260.03
***By the way, this is an example of why the way you're doing it reduces the Variance slightly. In the case of all twenty cards being literally the same five numbers, so really, just one card twenty times...if you hit all five numbers, then you win 838 * 20 * (Bet per Card), right? In the case of the way you're playing the game, that's essentially never going to happen. You'd have to hit your four numbers AND all sixteen others.
With that, if we assume that you hit the four spot, then anything from 0-11 five spots hit wouldn't be anything too crazy.
In addition to House Edge, Variance can also be a relevant concept, as we saw in this case. In the case of Four-Spot Keno, it would be highly unusual to still be ahead over very many sessions at all, but that’s because the Variance of that game is so much lower. With Five-Spot Keno, if you hit top results a few times, then you can be profitable for, potentially, a very long period of time.
Let’s close out the year with a nice easy question.
CONCLUSION
Video Keno is my favorite sub-100% Return-to-Player game; honestly, it’s the only one I like, so that seems like a good place to end the year.
I hope that you found this two part series of my best answers from 2023 informative. I’ll probably do this as a year-ender thing next year, as well, unless any readers want me to do this more regularly.
If you do want me to do it more regularly, what would really help with that is coming over to WizardofVegas, creating an account and creating a thread to ask your gambling questions. Feel free to send me a PM and I will generally get to them within 24 hours! If I got more excellent questions, then I would write more pretty good articles.
Merry Christmas and Happy New Year! I hope to see you and your questions in 2024!
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