STICKMAN’S STANCE: WHAT ARE THE ODDS OF FILLING THAT VIDEO POKER HAND? by Jerry "Stickman"
If you are at all like me, when you get a hand that has four of something – like four of a flush or four of an open straight – you notice when you do not complete the hand. It seems that these hands are filled much less often than they should be. One slightly eccentric video poker "expert" even claims you can tell if a video poker game is biased simply by observing the outcome of a few four-of-a-flush hands. I don’t put much stock in his claim. Randomness is strange. You may go 20 or 30 four-of-a-flush hands without filling a single one and then get five or six in a row. (We don’t tend to remember the latter.) This series of events takes considerable time to play out. To truly see the probability of random events, you must have a very large sample of events. Several four-of-a-flush hands over 30 minutes to an hour are simply not a large enough sample. A very good way to observe randomness in a video poker game is to play a 100-play game. The nice thing about 100-play is you can see 100 results of a given starting hand. If a particular hand should be filled 20 percent of the time, you should have 20 occurrences of the filled hand. There may be times you only have one or two occurrences, but there will also be times you have 25, 30, or more occurrences. It is much easier to see how the low-occurrence series are balanced out by the high-occurrence series in a 100-play game. I thought it would be interesting to look at the probability of completing several fairly common originally dealt hands. So let’s begin. All the examples that follow are for games with no wild cards. Probability of completing hands that contain four for the final hand Four of a flush: Let’s start with the hand that has been used by some as an indication of fairness of a game. Once the original hand of five cards is dealt, there are 47 cards left in the deck. Each suit has 13 cards, so there are nine of 13 cards of the flush suit still available. This means there is a nine-in-47 chance of filling the flush when dealt four. This works out to about 19 percent or about one in five. To average one in five, a player could go 20 hands without completing a single flush (which really seems like a long time), and then complete five in a row. Now the odds of getting exactly four of a flush are about three percent – three hands out of one hundred. If you play 800 hands per hour, you can expect 24 initial hands of four-of-a-flush. Of these 24, only 19 percent, or about four or five, should be completed. It does not fall very far from the norm to get only one or two flushes and no flushes is not outside the realm of randomness. Four of an Open Straight: An open straight can be filled by getting a card on the low end or a card on the high end of the four-card straight. There are four cards on the low end and four cards on the high end that will complete the straight for a total of eight out of the remaining 47 cards. This works out to 18 percent – slightly less than one in five. While slightly less likely to complete than four of a flush, it is still not bad. Again, just like four of a flush, completing four of an open straight might not happen for dozens of hands only to be followed by a few quick hits to bring the completion rate back to near mathematical odds.
Four of an Inside Straight: An inside straight can only be completed by filling a gap in the straight. That means there are only four cards of the remaining 47 that will complete the straight. The odds of completing this hand are 4-out-of-47 or about 8.5 percent – about one in eleven. This low completion rate is why four of an inside straight is normally only saved if there are three high cards in the four.
Four of an Open Straight Flush: An open straight flush is four of an inside straight that are also all the same suit. Getting this hand can start the adrenaline flowing as a straight flush is the second highest paying hand in Jacks or Better, and within the top several best pays in most other games. What are the chances of completing a straight flush when dealt four without a gap? There are only two cards in the remaining 47 that will complete a straight flush – the next higher card of the same suit in the sequence or the previous lower card of the same suit in the sequence. Therefore, the odds of completing a straight flush when dealt four in sequence are two-out-of-47 or about four percent – not very good odds. In a hundred-play game there would be about four completions when dealt four on an open straight flush. A player could go a very long time without ever completing four of an open straight flush in a single-play game. Four of and Inside Straight Flush: Being dealt four of an inside straight flush can also get some excitement flowing because it normally pays 50-for-1. However, because it is an inside straight flush, completing the hand requires the single card in the remaining 47 that fills the gap – a one-in-47 chance of occurring. A player on a 100 play game that is dealt four of an inside straight flush will only complete the straight flush twice in the 100 hands – on average. Of course, there would also be several flushes and a few straights also thrown in to help in the win column.
Three of a Kind: Being dealt three-of-a-kind can be a very exciting occasion because four-of-a-kind always pays well and sometimes pays extremely well – as much as a royal flush on some pay tables. However, what are the chances of completing the four-of-a-kind? The bad news – there is exactly one card in the 47 that remain the will complete the hand – the same as with an inside straight or an inside straight flush. The good news? There are two chances to get that scarce card. That works out to about four percent – or just better than one in 25. Playing a 100-play game, there would be about four four-of-a-kind completions in a round on average. As with other possible hands, these completions could come after 100, 200 or more hands without a single four-of-a-kind before snagging a few quads in close proximity. This is the nature of randomness. A dealt three-of-a-kind is still a great hand as it is a multiple coin winner in itself plus the possibility of a full house or a four-of-a-kind. Four of a Royal Flush: This is the ultimate non-winning dealt hand. It is only one card away from the top jackpot in video poker – so powerful, it is preferred to any hand except four-of-a-kind, straight flush, or a royal flush. Even the most experienced video poker players tend of pause a moment after selecting the four powerful cards before hitting the draw button. A royal flush is simply the highest possible straight flush, so four of a straight flush is the same as four of an inside straight flush as only one card in the remaining 47 will complete it. Therefore, the odds of completing four of a royal flush are one-in-47 – the same as completing any inside straight flush. A player lucky enough to be dealt four of a royal on a 100-play game would expect to complete two royals in the round. Again, just because the average is about two times out of 100 hands, does not mean it happens all the time. There could be hundreds of hands before hitting the first royal flush and then there could be a couple hits in a fairly short amount of time. The odds of being dealt four of a royal flush are fairly remote – only about one in 2,765. Add to this the one-in-47 chance of actually completing the royal and a player averaging 800 hands an hour will be dealt four of a royal only once every 3.5 hours on average. If you factor in the effects of randomness, it could be dozens of hours of play before a four-of-a-royal-flush hand is dealt. No wonder this hand is so exciting. Not only does it provide the best opportunity for the top jackpot, but it also occurs so infrequently. The next time you are tempted to think you have really bad luck, remember good hands do not appear very often, and completing these good hands is infrequent. When combined with the quirks of randomness, it can be what seems like an interminably long time between completions. Until next time…. All the best in all your casino and life endeavors. Jerry "Stickman"
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