The odds of drawing a straight flush from the initial deal in a five-card poker hand are one in 72,193.33. Those odds jump to one in 649,740 for a royal flush.
Last changed 1/9/99
Quite often, people use the well-known normal distribution toestimate the kind of results that you should expect whenplaying video poker. But, this estimate is off by quite abit unless you are talking about a very largenumber of hands (around 4,000,000 should do). For a smallernumber of hands, we need to be more careful. On this page,I will examine four popular video poker games with quitedifferent characteritics. Note that variance is measuredin units of bets squared.
Video poker games tend to have high variance, and most of that varianceis carried in the high payoff hands (typically Royal flush and specialbonus hands such as Four Deuces).Although these hands have a majoreffect on the EV, for short term play, the probabilities of the common hands are more important. The followingchart shows the probability distribution of ending bankrolls afterplaying 1000 hands of the four games specified.
Royal Flush Odds In Video Poker. Royal Flush is, hands down, the most sought-after hand in video poker. It consists of five highest-paying cards, from 10 to Ace. A lucrative hand like that is a dream of any gambler. The sad thing is that a Royal Flush is a very rare hand and your chances. The bottom line is as long as the pay schedule is the same for a particular video poker game, the odds of getting a royal flush will be the same no matter where the machine is located (assuming a random deal). As you said, the chances of being dealt a royal flush are 4 out of 2,598,960 or 1 out of 649,740. It's harder to figure out the chances of drawing to a royal flush. It's relatively easy to figure out the chances when you hold 1 card to the royal, 2 cards, 3 cards, 4 cards and 0 cards. But that's only half the problem.
Although the area under each curve is the same (they all integrateto give 1.0), they have different shapes. The Jacks or Bettercurve is sharpest because JB does not have a second high payinghand besides the Royal. On the other hand, Deuces Wild hasa distinct 'knee' at around 200 units, which corresponds to the200 unit payout of the 'mini-jacpot' of four deuces.Also notice the little 'blips' at around 800 units on each curve.All of the games pay 800 for a Royal, and those blips representthe chance of getting a Royal in 1000 hands.
Looking at these curves alone, it is hard to see that Deuces Wildhas the highest EV and Jacks or Better the worst.In fact, for such a short session, you are more likely to benear even, and to avoid a large loss, play JB rather than the othergames, even though it has the worst EV.So, what happens if we look at a larger session?
Probabilities - 10,000 Hands
After 10,000 hands, things start to smooth out quite a bit.It is apparent that the lower EV of JB is starting to dragit downward -- the long term expectation is to lose yourbankroll playing any negative expectation game. Also noticethat the 'four deuces' knee has been completely absorbed inthe Deuces curve, and the Royal kneeis starting to merge for Jacks (itwill take about 30,000 more hands before the Jacks kneeis no longer apparent). Also, Deuces is starting to looka good bit better than Double Bonus (especially on theleft hand side, where the losses are!).
But, 10,000 hands may be a bit long for a session for mostplayers. I picked 5,000 hands as a good intermediate valuethat is close to what I expect most players would plan fora short trip.
Odds Of A Royal Flush In Video Poker Results
After 5000 hands, the left hand Jacks peak is almost perfectlynormal --- but it matches a normal curve with a variance (and expecation) corresponding to subtracting out the Royal.Because cases where a Royal does occur (represented by theright hand peak) are well-separated from the first case, here we can get a good approximation of the distributionby combining two normal curves. But that is not truefor Deuces or Double Bonus (Pickem is quite close to normalby this point because the very high paying hands are sorare). Notice that both Deuces and DB are not symmetric-- the right hand side of the curve has 'lifted its foot'considerably.
So, what does this mean to the player? I'm glad you asked.We can redraw the same curves to give the cumulative probabilitiesand read off what to expect on a session of 5000 hands.
Cumulative Probabilities - 5000 Hands
For any given ending bankroll (starting from zero, remember), thecurves indicate the percentage of the time that you will end upat or below the indicated point after 5000 hands of the chosengame. Looking at the vertical line at zero ending bankroll, we cansee that you should expect to lose at Jacks or Betterabout 72% of the time, but only about 55-56% of the time at theother games.From the other angle, the 50% horizontal line show that halfthe time we should lose 102 units or more when playing JBfor 5000 hands, but only 32-51 (or more) for the other games.
From a trip bankroll perspective, the 5% and 10% lines maybe the most interesting.For a 5000 hand session, Double bonus can be cruel -- youwill end up losing over 400 units one trip in 10, and 488or more 1 in 20. The other games aren't quite so bad, butyou must be prepared for the variance.
Be careful of one thing --- the curves assume you play thefull 5000 hands. So, you might dip below the ending bankrollon the way to getting there. You really need a bit more thanthe curves indicate (ballpark estimate: 10%) to be sure you don'tend up with playing more before the end of your trip morefrequently than you had planned for.
I hope these curves will help you to understand the cruelyet enticing game of video poker. We can't get rid of thevariance, but we can understand and prepare for it.
--jazbo
The ins and outs of that most elusive of hands
By Henry Tamburin
I get many questions about a royal flush in video poker. That’s not too surprising since the royal flush is the premier hand that all video poker players dream (and hope) of getting. Here’s a sample of questions and my responses.
Q: I’ve been playing video poker several times a week for over a year. You keep saying that a royal flush occurs once in every 40,000 hands yet I still haven’t gotten a royal. What gives?
Odds Of Hitting Royal Flush
Firstly, I never wrote that you could expect one royal flush after playing 40,000 hands (or one cycle). What I wrote was, “On average, you will hit a royal flush once in every 40,000 hands.” The word “average” means a whole bunch of sets of 40,000 hands. In other words, in any given set of 40,000 hands, you could hit more than one royal flush or, heaven forbid, possibly no royals. In fact, you have a 36.8% chance that you won’t get a royal in one cycle (40,000 hands), and a 13.5% chance after two cycles (80,000 hands). Ouch! Therefore, the fact that you went over a year without a royal is statistically possible.
Q: How come every time I need one card for a royal flush, it never shows up, but that exact card that I needed always seems to show up on the very next hand?
That’s because you have “selective memory.” The computer program in the video poker machine that randomly selects the cards for each hand doesn’t use the information from previous hands to determine which cards it will deal. Every hand is a random deal regardless of what cards appeared (or didn’t appear) on the previous hand.
Q: Over three years, I hit seven royal flushes in the same casino and none in two other casinos that I play regularly. I’m beginning to believe those casinos somehow tighten their video poker machines so players can’t get a royal.
You will average one royal flush per roughly every 40,000 hands at any casino. Casinos can’t change the odds of hitting a royal flush. (What they can do is change the payout … some casinos will pay less than 4,000 coins for a royal flush; therefore, always check to be sure that the payout for a five-coin royal flush is 4000 coins.) The bottom line is as long as the pay schedule is the same for a particular video poker game, the odds of getting a royal flush will be the same no matter where the machine is located (assuming a random deal).
Q: I’ve been dealt many three- and four-card royal flushes lately. What are the odds of this happening?
Playing Jacks or Better, you’ll experience the thrill of being dealt a four-card royal flush once in every 2,777 hands (roughly once every four hours on average). Once in every 92 hands, on average, you’ll be dealt a three-card royal flush (about 7-8 per hour). This is what makes video poker exciting; namely, that you’ll have several opportunities to draw for a royal flush even if the odds are somewhat long (see next question).
Q: When you hold three cards to the royal flush, what is the chance of getting the two cards that you need on the draw for a royal flush?
You have a one in 1,081 chance of getting the two cards you need for the royal flush. The following table shows the chance of hitting the royal flush on the draw when you hold x cards to the royal flush.
Q: If I’m dealt a three-card royal flush and a high pair in the same hand, why does the strategy say to hold the high pair when the royal flush pays so much more?
You need to analyze all the possible winning hands that you could get when you hold a three-card royal flush vs. when you hold a high pair in the same hand. These calculations have already been done for you. For example, suppose your initial hand contains 10-J-Q of diamonds along with a queen of clubs. The expected return (ER) for holding the pair of queens is 7.6827 vs. 7.4098 for holding the three-card royal flush (this is for 9/6 Jacks or Better). In dollars and cents, you’d earn 27 cents more on average for a max coin wager on a dollar denomination machine by holding the high pair vs. the three-card royal flush in this example.
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Q: My wife plays Jacks or Better. The other day she was a dealt a hand that contained a four-card straight flush with a gap and a three-card royal flush. She held the three-card royal flush. Was that the correct play?
I’m sorry to say it wasn’t. The correct play was to hold the four-card straight flush—even with a gap—over the three-card royal flush. (Tip: If your wife had a strategy card with her, she would have made the right play.)
Q: What are the odds of being dealt a royal flush in the initial hand?
The odds are one in 649,740 hands. You might think that’s close to impossible but it could happen. (This happened to me once while I was showing my father-in-law how to play a Triple Line video poker game in a Las Vegas casino, resulting in a royal flush on each line. How’s that for luck?)
Q: How much does the royal flush contribute to the 99.54% return for 9/6 Jacks or Better?
The royal flush contributes 1.9807% toward the overall 99.64% return. The following table summarizes the contribution of each winning hand toward the overall 99.54% return (for 9/6 Jacks or Better). When you don’t hit the royal or straight flush, the best return you can expect, even playing perfectly, is about 97%.
Odds Of Getting A Royal Flush In Video Poker
Got a video poker question? Send it to [email protected].
Tamburin’s Tip of the Month
You are playing NSU Deuces Wild. How would you play these hands that don’t contain a deuce?
In the top hand, your best play is to hold the consecutive three-card straight flush 6-7-8 (2.77 ER) over the four card straight 5-6-7-8 (2.55 ER). In the bottom hand, because the three-card straight flush has a gap (2.47 ER) your best play is to hold the consecutive four-card straight 4-5-6-7. When you play NSU Deuces Wild and your initial hand doesn’t contain a deuce, you should hold a consecutive three-card straight flush (5-6-7 through 9-10-J) over a consecutive four-card straight (from 4-5-6-7 to 10-J-Q-K), but the latter over a three-card straight flush with one or two gaps.
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Henry Tamburin is a blackjack and video poker expert. He is the host of the smartgaming.com website and the editor of the Blackjack Insider newsletter (for a free three-month subscription, visit www.bjinsider.com/freetrial). For a free copy of his Casino Gambling Catalog, which contains books, strategy cards, and software for video poker players, call toll free 1-888-353-3234, or visit the web store at smartgaming.com.
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