# A simple probability question on the odds of getting a flush **URL:** \ **Category:** The Game Room\ **Created:** [January 11, 2013, 3:28pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865 "2013-01-11T15:28:24Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![dalej42](https://avatars.discourse-cdn.com/v4/letter/d/67e7ee/32.png) [@dalej42](https://boards.straightdope.com/u/dalej42)\ **Post date:** [January 11, 2013, 3:28pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/1 "2013-01-11T15:28:24Z")
I am playing video poker, there is one 52 card deck, the cards are dealt at random. I am dealt 4 cards of one suit and one card of another suit. The strategy guide, of course, tells me to discard the card of the other suit to try to try for a flush (5 cards of one suit.) Of course, it always seems like I end up not getting the flush! What are the odds of me drawing the flush? This is not homework, just my frustration with playing video poker. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [January 11, 2013, 3:32pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/2 "2013-01-11T15:32:01Z")
There are 47 cards left in the deck, (assuming that it was shuffled before this ‘deal’ and no other known cards have been dealt) 52 - 5 = 47 Assume your four-flush is in spades. There are nine spades left, (13 - 4 = 9) and thirty-eight cards left in the other suits. (39 cards for three suits, subtract the one non-spade you were dealt.) Thus, your chances of completing the flush are 9 in 47, or about 19% ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 11, 2013, 3:34pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/3 "2013-01-11T15:34:14Z")
Not an answer, but if the guide is correct, the odds of drawing a flush are better than the odds of discarding any number of your cards and getting a better hand. So we should also look at the odds of trying for a better hand. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [January 11, 2013, 3:57pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/4 "2013-01-11T15:57:09Z")
> [@TriPolar](#): > > Not an answer, but if the guide is correct, the odds of drawing a flush are better than the odds of discarding any number of your cards and getting a better hand. So we should also look at the odds of trying for a better hand. You don’t just need to improve your hand, you need to improve it enough so that it’s likely to beat the dealer. In other words, even if you’re more likely to get a lower-ranked hand using a different strategy, it doesn’t help you if it’s still a “below-average” hand. A flush is a darn good hand (there are only three types of hands more likely to win), so it’s more likely that you’ll beat the dealer if you end up with that hand than if you end up with (say) a pair. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [January 11, 2013, 3:59pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/5 "2013-01-11T15:59:17Z")
> [@MikeS](#): > > You don’t just need to improve your hand, you need to improve it enough so that it’s likely to beat the dealer. In other words, even if you’re more likely to get a lower-ranked hand using a different strategy, it doesn’t help you if it’s still a “below-average” hand. A flush is a darn good hand (there are only three types of hands more likely to win), so it’s more likely that you’ll beat the dealer if you end up with that hand than if you end up with (say) a pair. Nitpick - is (all) video poker ‘beat the dealer’ style? I’ve only played a few times, long ago, but I don’t really remember a dealer hand, I remember a payout system where getting different hands after the draw were worth different multiples of your original stake paid back to you. ---
**Author:** ![Procrustus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/procrustus/32/2994_2.png) [@Procrustus](https://boards.straightdope.com/u/Procrustus)\ **Post date:** [January 11, 2013, 4:00pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/6 "2013-01-11T16:00:49Z")
> [@MikeS](#): > > You don’t just need to improve your hand, you need to improve it enough so that it’s likely to beat the dealer. In other words, even if you’re more likely to get a lower-ranked hand using a different strategy, it doesn’t help you if it’s still a “below-average” hand. A flush is a darn good hand (there are only three types of hands more likely to win), so it’s more likely that you’ll beat the dealer if you end up with that hand than if you end up with (say) a pair. The OP is playing video poker. Hence, he/she has to get Jacks or better, there is no dealer hand. Unlike the OP, I am surprised how often I seem to get a flush in this situation. I love getting dealt four of one suit. ---
**Author:** ![hogarth](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hogarth/32/1773_2.png) [@hogarth](https://boards.straightdope.com/u/hogarth)\ **Post date:** [January 11, 2013, 4:03pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/7 "2013-01-11T16:03:24Z")
I was under the impression that video poker deals weren’t necessarily uniformly distributed, but the machine had to list the odds of each particular hand somewhere. Am I just confused? (Probably.) ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 11, 2013, 4:06pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/8 "2013-01-11T16:06:04Z")
> [@MikeS](#): > > You don’t just need to improve your hand, you need to improve it enough so that it’s likely to beat the dealer. In other words, even if you’re more likely to get a lower-ranked hand using a different strategy, it doesn’t help you if it’s still a “below-average” hand. A flush is a darn good hand (there are only three types of hands more likely to win), so it’s more likely that you’ll beat the dealer if you end up with that hand than if you end up with (say) a pair. Ok, I don’t play video poker. Are you saying that you know the dealer’s hand (or some part of it) before drawing? And yes, if that’s the case, the odds of beating the dealers hand is what’s important. ---
**Author:** ![dalej42](https://avatars.discourse-cdn.com/v4/letter/d/67e7ee/32.png) [@dalej42](https://boards.straightdope.com/u/dalej42)\ **Post date:** [January 11, 2013, 4:22pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/9 "2013-01-11T16:22:56Z")
There is no dealer hand in video poker. You need to get a hand that matches one of the hands on a pay schedule. So, if I have Ace, 3, 5,7 of Spades and a 9 of hearts, I win nothing. If I discard the 9 of hearts and get a 10 of spades, I will win a certain number of coins displayed on the pay table. The selection of the cards dealt is determined by a random number generator. ---
**Author:** ![dalej42](https://avatars.discourse-cdn.com/v4/letter/d/67e7ee/32.png) [@dalej42](https://boards.straightdope.com/u/dalej42)\ **Post date:** [January 11, 2013, 4:27pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/10 "2013-01-11T16:27:53Z")
Information on how a video poker machine works from [videopoker.com](http://videopoker.com) [http://www.videopoker.com/learn/superstitions/#rng](http://www.videopoker.com/learn/superstitions/#rng) …The RNG is the hardest working device known, and it ensures a fair game every time. Its sole responsibility is to constantly shuffle the deck of 52 cards (or 53 in Joker Poker). This manic activity goes on and on UNTIL you hit the “deal” button. When you hit the deal button, the machine displays the 5 cards at the top of the deck at that precise moment… Then it keeps on shuffling the remaining 47 cards (or 48 in Joker Poker) until you hit the deal button and the discarded slots are filled with whatever is at the top of the deck at that time. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [January 11, 2013, 4:28pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/11 "2013-01-11T16:28:01Z")
> [@dalej42](#): > > There is no dealer hand in video poker. You need to get a hand that matches one of the hands on a pay schedule. So, if I have Ace, 3, 5,7 of Spades and a 9 of hearts, I win nothing. If I discard the 9 of hearts and get a 10 of spades, I will win a certain number of coins displayed on the pay table. > > The selection of the cards dealt is determined by a random number generator. That’s what I thought you were talking about. So in terms of strategy you need to know the odds of getting a better hand by drawing more than one card. If you draw one card all you can do is improve your hand from nothing to a pair, or a flush (or a straight flush is you have the four cards in series already, or a series with a one card gap). ---
**Author:** ![bup](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@bup](https://boards.straightdope.com/u/bup)\ **Post date:** [January 11, 2013, 4:29pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/12 "2013-01-11T16:29:15Z")
So if the payout is greater than 100/19 of your initial bet (based on the 19% figure above), you should be happy to get dealt 4 of one suit. Even if not, it’s still probably your best bet, unless the one offsuit card makes a pair. Assuming you don’t have a pair, the next best hand would be to have 2 or 3 of your cards be jacks or better, and to discard two (or three) to try to pair them up. Then you’re trying for 9 (or 6) of 47 cards left in the deck, with 2 (or 3) chances. The odds are worse than 19%. ---
**Author:** ![Procrustus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/procrustus/32/2994_2.png) [@Procrustus](https://boards.straightdope.com/u/Procrustus)\ **Post date:** [January 11, 2013, 4:43pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/13 "2013-01-11T16:43:33Z")
Where it gets interesting is if you have four of one suit, but a high pair like two Aces. So, you have a hand that is already at least a push, and you can draw three to try to make 3 or 4 of a kind, or two pair, or full house. Or you discard your Ace and try for the Flush, knowing more times than not you won’t make it. My head can’t do the math on that, but I’d probably keep the high pair. ---
**Author:** ![silenus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/silenus/32/166_2.png) [@silenus](https://boards.straightdope.com/u/silenus)\ **Post date:** [January 11, 2013, 5:02pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/14 "2013-01-11T17:02:46Z")
The strategy cards always say dump the odd card and go for the flush rather than go for a 3-of-a-kind. The difference in payoff amounts over the long run make going for the flush the higher-paying option. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [January 11, 2013, 5:22pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/15 "2013-01-11T17:22:07Z")
Ah, mea culpa; I was somehow thinking that video poker was like blackjack. If there are listed payouts, then the optimal strategy would instead be determined by the relative net payouts of each type of hand (as alluded to by **silenus** ). For example, if I have a 19% chance of getting a flush with one strategy, and a 38% chance of getting three of a kind with another strategy, I should still pick the first strategy so long as the net payout for the flush is more than double the net payout for three of a kind. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [January 11, 2013, 5:23pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/16 "2013-01-11T17:23:25Z")
> [@dalej42](#): > > There is no dealer hand in video poker. You need to get a hand that matches one of the hands on a pay schedule. So, if I have Ace, 3, 5,7 of Spades and a 9 of hearts, I win nothing. If I discard the 9 of hearts and get a 10 of spades, I will win a certain number of coins displayed on the pay table. Nitpick: AFAIK, with most video poker, a pair that’s less than Jacks counts the same as nothing at all. So drawing that 10 of spades won’t help you. ---
**Author:** ![yearofglad](https://avatars.discourse-cdn.com/v4/letter/y/b782af/32.png) [@yearofglad](https://boards.straightdope.com/u/yearofglad)\ **Post date:** [January 11, 2013, 5:27pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/17 "2013-01-11T17:27:22Z")
> [@CurtC](#): > > Nitpick: AFAIK, with most video poker, a pair that’s less than Jacks counts the same as nothing at all. So drawing that 10 of spades won’t help you. Except that with the Ace, 3, 5, and 7 of spades still in hand, the 10 of spades will make a flush. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [January 11, 2013, 5:34pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/18 "2013-01-11T17:34:39Z")
> [@MikeS](#): > > Ah, mea culpa; I was somehow thinking that video poker was like blackjack. If there are listed payouts, then the optimal strategy would instead be determined by the relative net payouts of each type of hand (as alluded to by **silenus** ). For example, if I have a 19% chance of getting a flush with one strategy, and a 38% chance of getting three of a kind with another strategy, I should still pick the first strategy so long as the net payout for the flush is more than double the net payout for three of a kind. Yeah. And some strategies may give you odds for different payouts - such as two pair or three of a kind, drawing to a pair. ---
**Author:** ![CC](https://avatars.discourse-cdn.com/v4/letter/c/f1d935/32.png) [@CC](https://boards.straightdope.com/u/CC)\ **Post date:** [January 11, 2013, 5:35pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/19 "2013-01-11T17:35:43Z")
Strategy in video jacks or better poker is based not only on the odds of improving your hand, but also on the nature of the payout, which actually differs depending upon the particular machine you are playing on. There are a few different ways to calculate the best strategy and they don’t all agree on all points. Mensa’s guide tells you that you should keep your four card flush unless you’re holding these hands, in which case you play them instead of the 4-flush: Royal flush Straight flush four of a kind four card royal flush full house flush three of a kind straight four card straight flush (a four flush, obviously) two pair pair of jacks or better three card royal flush p.s. I believe that this chart pertains to betting the maximum bet all the time, meaning that a royal flush will pay off at a much higher rate than it does for lesser bets. ---
**Author:** ![Colibri](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colibri/32/1841_2.png) [@Colibri](https://boards.straightdope.com/u/Colibri)\ **Post date:** [January 11, 2013, 6:02pm UTC](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865/20 "2013-01-11T18:02:02Z")
Moved to the Game Room. Colibri General Questions Moderator [Next page](https://boards.straightdope.com/t/a-simple-probability-question-on-the-odds-of-getting-a-flush/646865.md?page=2)