For a greater overall expected profit, I recommend using a less aggressive royal strategy to increase the total hands played. Normally the return is 1.00762 and a royal hits once every 45282 hands. Let’s look at full pay deuces wild, for example. When I moved to Vegas three years ago it was much easier to find. I think all the Stations and Fiesta casinos have full pay deuces wild in the quarter coinage only. This formula can be used for any game by substituting the correct standard deviation per hand. In the case of my first example you would put the following in any cell: 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. This is still a lot, which just goes to show how volatile your bankroll will be in the short run.

That is a lot! Suppose your goal is to simply come out ahead with a 90% probability.

I’ll skip over the rest of the math to the answer, which is 42,383,720 hands. The standard deviation of the sample mean is 5.08/sqr(n), where n is the number of hands. The standard deviation of full pay deuces wild is 5.08. Perhaps a better question is how many hands would I have to play to have a 90% probability of getting to 100.66%? 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. There is no number that will give you a 90% probability of attaining or exceeding that percentage. So in that situation the player should be mindful of the effect of penalty cards.' 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. However this is only marginally better than discarding everything. 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. By discarding it the players odds of forming a flush are 'penalized.' Sometimes penalty cards can affect borderline plays. The discarded suited card would be called a flush penalty card because it could have been used to complete a flush. 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. However for those who must play near perfectly here is my explanation, 'A penalty card is a potentially useful discarded card. In my opinion beginning players should not use a strategy that considers penalty cards. I just rewrote my explanation of penalty cards.

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