For the talented mathematical minds of the SDMB and anyone interested, I’m trying to calculate the odds of sucessfully “doubling” in video poker. I’m sure most people who play video poker will be familiar with this but since I’ve never encountered the option to “double” in computer video poker games and there can be several different methods I think a brief explanation is in order.
The machine is Jacks or Better, although that isn’t really relevant. When you hit a winning hand, the machine gives you the option to try to double your winnings. Using a 52 card deck in random order (what do you call it in english? Mixed deck? Unsorted deck? I dunno…) the machine draws a card (2 being the smallest and ace the biggest). Then you’re given four cards (randomly out of the 51 left) facing down and choose one of them. If it’s the same card as the machine’s (the suit isn’t relevant), it’s a draw (you can try all over again if you wish), if it’s inferior you lose all your winnings, if it’s superior you double your winnings (and you can keep doubling until you hit the machine’s limit).
My question is what’s the probability of doubling your winnings (before you know the dealers card, obviously, because that’s when you get to choose to double or not)?
If we know this probability, let’s call it p, then the probability of doubling n times in a row is simply p[sup]n[/sup], so that’s not a problem.
Without getting into too much detail for now, I did some calculations and arrived at a figure of approximately 45% to win or draw (i.e. not lose), which agrees with my playing experience (40%-50%) and makes sense because there is no way the casino is going to give odds better than even.
Problem is, I’m at loss to explain why this doubling method favours the house. In fact I suppose it should be the opposite, since there are 7 draws (2 through 8) for the machine that favour the player and only 6 (9 through ace) that favour the house.
I hope I was clear, any thoughts on this are welcome.
I believe the doubling calculations you’re doing are flawed regarding the 8s. When an 8 is drawn there are 24 cards in the deck which will be higher, and 24 which are lower, and the other 3 to tie so the 8 does not favor either the player or the house, and so the overall chance is 50% with this doubling.
In the video poker machines developed by the company I work for, doubling is 50% chance and to go for 4x your win is 25% chance, and doesn’t work like this - you guess the color (for double) or suit (for 4x) before one card is selected.
You’re correct AmbushBug, thanks. I called the three 8s that draw as favouring the player, which is not true. But also don’t forget that I defined “success” as winning or drawing. Considering that, it still doesn’t explain why the method favours the house, if indeed it does.
By the way, slight correction, I detected a mistake and I got p = 27/51 = 0.5294 so now I’m convinced my calculations are completely wrong. I’ll post them later.
One other thing, don’t be fooled by the part where is says to “choose between 4 cards facing down”. This is just designed to make the player feel like he has some control (it’s kind of fun indeed). But since the deck where those cards are taken from is randomly ordered, it’s just like picking the top card of the 51 left after the machine draws. The act of choosing out of four is inconsequential, every card is equaly likely (again, out of the 51 left).
I’m reminded of another thing, please bear with me.
I’m not sure if the machine (I’ll just call it dealer) uses one deck to draw for itself and a different deck to draw the four cards that the player gets to choose, or if it’s all done from the same deck like I’ve been saying. I tried asking the casino staff but they’re all clueless or they pretend to be.
Well if you define “success” as either winning or drawing, then “success” will happen over 50% of the time (52.94% of the time, 351/663 summed for all combinations which indeed reduces to 27/51). You can therefore “succeed” without having your win doubled, as in a tie. No matter how you count the ties your chance of doubling is a stone 50%.
Of the 663 possible draw combinations, 312 are losers, 312 are winners, and 39 are draws.
Why would a casino enable doubling if there’s no house advantage to doing so? Because some players like doubling, and anything which keeps the player at the machines is what makes the money.
You’re right, it’s a fifty-fifty chance to double or lose, I was making this more complicated than it need be.
By the way do you know off the top of your head the expected payout of a Jacks or Better machine? Specifically what’s called, I believe, a 7/5 Jacks or Better? I’ve picked up somewhere that Jacks or Better machines usually have the highest payouts but the 7/5 could be as low as 92%. Figuring my non-optimal play, maybe 85%.
I don’t have this sort of data at work, but this has some believable numbers.
Thing to remember about video poker is that there’s an element of skill involved (plus always playing maximum coin) in achieving the “perfect play” to get that return.
I’d appreciate correction, if I am mistaken, but it seems to me that with a fair draw, you can make the calculation more intuitive by noting:
This is equivalent to asking: "if I choose two card from a randomized deck, are the odds of the A>B equal to B>A?
The answer is “Yes, they are equal”. It doesn’t matter if I preordain drawing the 1st or Nth card from either deck. This is implicit in the term “randomized”.
My error, if any, would probably lie in my assumptions, not my math. Pre-choosing to draw card X or Y from a randomized deck does not give any additional information, compared to simply drawing the top card, and the a priori odds of a given card being in position X or Y are equal in a randomized deck, by definition.
All of this assumes, of course, that what the video poker machine does is indeed equivalent to honestly shuffling an actual deck of cards and then letting you choose a random card from that deck.
It has been demonstrated that at least some digital fruit machines cheat on double-or-nothing: the outcome of the bet is simply determined in advance, and the game is then rigged to produce that outcome. If that is the case here as well, then trying to perform a probability analysis on the game is meaningless; the only way to calculate your chance of winning would be to look at the source-code of the machine’s software.
Coincidently I found that excellent page about casino gaming yesternight and spent some time exploring it. It gives 96.1% for the 7/5 versus 99.5% for Full Pay. Still, higher than I thought. Unfortunately the casino I go to doesn’t have any competition so it can afford to give the payouts it pleases and still make a filthy amount of money.
By the way you rock AmbushBug! Thanks for not letting my thread sink to the bottom like a rock.
Exactly so KP. I concluded as much after I did the math using conditional probability. Like you infer, you can discard entirely the cards that produce a draw. It doesn’t even matter wether the draws are from the same deck or from different decks, it still fifty-fifty, so that caveat is moot.
“Shuffling”, that’s the damn word I couldn’t remember on the OP if my life depended on it. Thank you for allowing me to sleep better at night.
What you say is true of course but I do think the video poker machines are fair. They must be by law and the casino is reputable. The have something like 1200 slot machines on the floor. They don’t need to cheat to make a lot of money.
That’s an interesting site about “fruit machines” there Martin. I don’t know where Pedro does his wagering but in every legal gaming jurisdiction I know of in the U.S. and Canada slot machines are generally not approved for use if found to be deceptive in any way. Certain jurisdictions have their own rules, written and unwritten, aimed at protecting the players (and the casinos).
Source code for games is provided to jurisdictions and their authorized testing divisions or their contracted outside testing firms. No game program (not even an upgrade!) can be placed on a casino floor without being approved for use in that casino’s jurisdiction.
Note that the site I linked to is specific to the UK, and in their FAQ and campaign goals they specifically point out that US fruit machines behave differently because of different laws. If Wikipedia is to be believed, all American jurisdictions, with the possible exception of Indian tribes, state that if a video game features cards or dice then they must behave like the real thing.
