Ask the Wizard: Deuces Wild video poker

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I recently had a tremendously lucky streak going with the deuces wild video poker game. I was in Las Vegas, and over the course of the weekend hit three natural royal flushes. I'm rounding here, so let's say I played 10,000 hands during the weekend. What were/are my odds of hitting this/again? Thanks so much for all of your insight! – Kevin from Long Island, New York

The Poisson distribution can be used to answer this kind of question. The general formula is e^-m*m^x/x!, where x is the number of the event you observed, and m is the expected number. In this case x is 3. The probability of a royal flush in "Not so Ugly Ducks deuces wild" is 0.000023. So the expected number in 10,000 hands would be 0.23. Thus the probability of hitting exactly three royals in 10,000 hands is e^-0.23*0.23³/3! = 0.161%. The formula in Excel for this is poisson(3,0.23,0). December 13, 2006

I've been playing 100-hand bonus deuces wild video poker at Bodog. When I have no 2s in my hand but an inside straight possibility you say to hold the inside straight chance. Typically I'll end up with maybe 25 coins back. When playing hands with garbage in them I've thrown them away and often I've ended up with at least that much if not better. So I was wondering, should play be modified a bit when playing 100 hands at once? Or is it actually something with the pay table that causes that one play not to be quite right? – Steven from Cary

I see Bodog follows the 1-2-3-4-4-9-15-25-200-800 pay table, which is known as “Ugly Ducks” and returns 99.42% with optimal strategy. Although I don’t indicate an Ugly Ducks strategy on my site, my Not so Ugly Ducks strategy should do quite nicely, and is indeed accurate for this play. To answer your question, the strategy depends only on the pay table. The number of hands makes no difference. What is right for one hand is right for 100 hands. On average you should get about 34 coins back playing the inside straight and 32 coins back tossing everything. However, actual results will vary. I would say you’ve just been unlucky with the inside straights if you have only been getting back 25 on average. June 23, 2006

I was playing a triple-play deuces wild machine and was delighted to be dealt four deuces. I kept the deuces and discarded the queen and hit the draw button. Of course I was paid accordingly, but the stranger next to me freaked out saying that I should have held the queen instead of drawing a new card. He cited the fact that malfunctions void all payouts. In future situations, should I really be concerned about such malfunction possibilities? – Eric from Las Vegas

No. Malfunctions in any video based game are extremely rare. In slots with moving parts they are more likely, but still on the order of one in a million. In video poker, mid-play malfunctions are virtually unheard of. The reason usually given for holding all five cards with four deuces is that otherwise you might hold the wrong cards by pressing the buttons incorrectly. In my opinion the probability of this kind of human error is much greater than the probability of a malfunction. June 9, 2006

Hi, first of all I woulld like to congratulate you for your great site. Very interesting indeed. I have a question regarding your "optimal" strategy in Deuces Wild. In your appendixes about this game you say that you should keep two suited cards or discard everything depending on what you have to discard. In the example you give in appendix 1 the hand is K

, Q

, 3♣, 4

, 7

. In this case everything should be discarded. Now if you get the same hand with an 8d as last card (instead of the 7d),then you should keep the suited Kh and Qh. My question is what difference does it make? Whatever you discard (3c,4d,7

or 3c,4

) the odds for getting a royal flush (or straight flush or even anything else) seem the same for me. Since you already have two cards for the Royal Flush why not keep them whatever you have to discard? - Alain P. from Paris

Thanks. I've answered penalty card questions before but because it is an important concept I'll do it again. You are right that the expected value keeping the king and queen is the same in both hands. However, the value of tossing everything is a little bit more with the 7d than the 8d. To be exact, if you toss everything with the 7

there are 1,606 ways to make a straight, and with the 8d there are only 1,591. The more spread apart your discards are, the harder it will be to make a straight on the draw. This particular case is very borderline. Those 15 extra ways to make a straight by tossing the 7

pushes the value of tossing everything over that of keeping two to a royal. For the same reason, in a six-deck game of blackjack you double A2 against a 5 but not a 4. The value of doubling is the same in both cases, but the value of hitting is more against the 4, pushing it above the value of doubling. May 10, 2006

On a full pay deuces wild machine how does a progressive jackpot affect the percent payback. For example on a full pay non-progressive machine the pay out for a Royal Flush no deuces is 4000 coins. This machine has a 100.76 pay out. How is the pay out percentage affected if the Royal Pays 4400 coins?

Assuming no strategy changes for every extra 100 coins in the jackpot the return goes up by 0.044%. So the return with a 4400 coin jackpot would be 100.762% + 4*0.044% = 100.939%. Sept. 18, 2005

What is the probability of playing 14,000 hands of deuces wild without getting four deuces?

We can see from my deuces wild analysis that the probability of four deuces in any one hand is 0.000204. So the probability of not getting four deuces in any one hand is 1-0.000204 = 0.999796. The probability of going 14000 hands without four deuces is 0.999796¹⁴⁰⁰⁰ = 5.75%. Dec. 5, 2004

I see that the return on a full pay deuces wild video poker machine is 100.76% with the strategy you have on your website. Obviously, this is with infinite play. My question is how many hands would you have to play to get that return with, let's say 90% certainty?

p.s. Where can I find full pay full pay deuces wild in Vegas?

There is no number that will give you a 90% probability of attaining or exceeding that percentage. Although the more you play the closer your return will get to 100.76% about half the time it will be above and half the time it will be below. Perhaps a better question is how many hands would I have to play to have a 90% probability of getting to 100.66%?
The standard deviation of full pay deuces wild is 5.08. The standard deviation of the sample mean is 5.08/sqr(n), where n is the number of hands. I'll skip over the rest of the math to the answer, which is 42,383,720 hands. That is a lot! Suppose your goal is to simply come out ahead with a 90% probability. Then you would only need 733,790 hands. This is still a lot, which just goes to show how volatile your bankroll will be in the short run. The general formula in Excel of your sample return being at least x is with a probability of p is (normsinv(1-p)*5.08/(1.0076-x))^2. In the case of my first example you would put the following in any cell:

=(NORMSINV(1-0.9)*5.08/(1.0076-1.0066))^2

This formula can be used for any game by substituting the correct standard deviation per hand.

I think all the Stations and Fiesta casinos have full pay deuces wild in the quarter coinage only. Green Valley Ranch too. When I moved to Vegas three years ago it was much easier to find. March 6, 2004

Playing deuces wild if I hold three deuces what is my probability of getting four deuces on the draw? What if I hold two deuces?

If you hold three deuces there are 46 ways you can get the other deuce and another card. There are combin(47,2)=1081 to choose two cards out of 47 left in the deck. So the probability of getting four deuces on the draw with three held is 46/1081 = 4.26% = 1 in 23.5. If you hold two deuces there are 45 ways to get two more deuces plus another card. There are combin(47,3)=16215 ways to choose 3 cards out of 47. So the probability of getting four deuces on the draw after holding two is 45/16215 = 0.28% = 1 in 360.33. Dec. 2, 2003

I have something rather interesting to calculate. I was playing Deuces Wile Video Poker when I was dealt "Garbage". When I threw all five cards away, I was given 4 deuces on the draw, 1,000 coins! What is the probability for 4-Deuces to appear on the draw after throwing all 5 cards away on the deal? Thanks for your time and keep up the good work on your website! - Saucedo, Anthony

P.S. On the same machine, I switched to Deuces Wild Bonus Poker and was dealt 3 wilds w/4&5 of diamonds (straight flush), I threw away the 4&5 and hit 4 deuces w/Ace for another 2,000! What a lucky machine! This was at Soboba Casino in Southern California.

No problem. There are combin(47,5)=1533939 ways to arrange 5 cards out of the 47 left in the deck. 43 of those will result in four deuces (the 5th card has 47-4=43 possibilities). So the probability of getting four deuces on the draw is 43/1533939 = 1/35673 = 0.000028032. The probability of drawing a fourth deuce after keeping 3 is (47-1)/combin(47,2) = 46/1081 = 0.0426. In my video poker appendix 5 you can see how the probability distribution in deuces wild of the number of cards drawn to any given hand. For example 2.62% of all four deuces will be obtained by getting all four on the draw. The probability of this happening on any given hand is 0.000005. For more information on the combin function visit my section on probabilities in poker. March 21, 2003

This is an excellent site very interesting and informative. When going through the list of plays in deuces wild video poker I didn't understand what is meant by a penalty card. Would you please explain. - Steve from Parma, Ohio

Thanks for the compliment. I just rewrote my explanation of penalty cards. In my opinion beginning players should not use a strategy that considers penalty cards. However for those who must play near perfectly here is my explanation, "A penalty card is a potentially useful discarded card. For example if the player had 3 to a royal and 4 to a flush the correct play is to keep three to the royal, discarding the fourth suited card. The discarded suited card would be called a flush penalty card because it could have been used to complete a flush. By discarding it the players odds of forming a flush are "penalized." Sometimes penalty cards can affect borderline plays. For example if the player had a suited 10 and king, with no other cards of that suit, nor a 9, jack, queen, or ace, then the player should keep the two to a royal flush. However this is only marginally better than discarding everything. If the player had just one suited card, or any card that could be used to complete a straight, then the odds of forming a flush or straight would be depressed, lowering the overall expected return below that of discarding everything. So in that situation the player should be mindful of the effect of penalty cards." Jan. 2, 2002

I was running your deuces wild video poker program. I had the following cards. QJ8 of spades T7 of diamonds. I opted for the 4 card inside straight draw and kept the QJT8. The advice thing came back and recommended that I keep the JT87 and discard the Q. This looks like one of those situations where there are two plays with the same EV (similar to the case where you have two pair and need to keep just one pair). - Dan from Albany, New York

You're right. There are two ways you can go for an inside straight, both of which have the same expected value. Sorry my program scolded you, I should correct that.

Are there any internet casinos with looses dueces wild and which casino has the best payouts for dueces wild. Thank you. - Mike from Montana

Atlantic Interbet has full pay deuces wild (return of 100.77%). I think their highest coinage is 50 cents in that game. April 15, 2001

How should I play the following hands in deuces wild video poker if both a flush and four of a kind pay 4 to 1? (1) two pair, (2) deuce and two suited high cards - Marvin

I'm going to assume the rest of the pay table is the same as full pay deuces wild. In this case (1) keep the two pair, (2) keep the deuce and both high cards. Feb. 18, 2001

I noticed that on your Deuces Wild strategy table that one pair (no deuce) is higher on the list that two pair. Does this mean that if I am dealt two pair, that I should always discard 3 cards and keep only one of the pairs? If so, how do I decide which pair to keep? - Dean from Toronto, Canada

Yes, you should only keep one of the pairs. The only exception to this two pair rule is if you also had three to a royal flush. It does not matter which pair you keep. This is one of the few situations among all games of skill where the optimal strategy player has some free will. Personally I always keep the higher pair, just so I don't slow myself down trying to decide. Dec. 24, 2000

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