Deuces Wild probability question

[olds442jetaway]

I really have to re learn these formulas, so if anyone feels like posting this one, thanks in advance. What is the probability of not hitting 4 Deuces on a regular Deuces Wild game 25/15/9/4/4/3/2/1 in 50k hands playing a quarter game with 5 coins in and flat betting? Also it would be for a single line game.

[alpax]

If you played perfectly on average it will be 1 in 5,347.73 to get quad deuces on that paytable

http://wizardofodds.com/games/video-pok ... 200-d-800/

50,000 hands / 5,347.73 = 9.3498 cycles

euler constant ^ -9.3498 = 0.00008698616

Otherwise 1 in 11496 chances you do not get quad deuces after 50,000 perfect play hands with max bet.

[olds442jetaway]

Thanks very much. Helps a lot in planning my Martingale clone play at some point on that game.

[Vman96]

Alpax's method is close enough for big samples, but it's not exact. And modern calculators can also handle the exact method for large samples...

Let p = probability of making four deuces in a given hand
let N = total number of hands played
Let P(0) = probability of not making four deuces at all in N hands

then..

P(0) = (1-p)^N


In the case of 50,000 hands of Airport Deuces:
1 - p = 1 - 1/5347.73 = 0.99981300

(0.99981300)^50000 = 0.000086910146 = 1 in 11,506

Edit: I misread your probability number.

[alpax]

Both formulas should lead up to the same result, I took the example from Wizard of Odds website, I remember reading about Royal Flush cycles on the FAQ about video poker. I switched out the numbers that Shackleford used.

The discrepency is that the true odds for Quad Deuces were not being used.

3,727,422,492 combinations out of 19,933,230,517,200 total turned out to be 5,347.72502 rather than 5,347.73 or 5,347.43. The small change in the decimal value changes the result!

[Vman96]

When trying to be anal and precise, it's pretty bad to misread a number...whoops.

But we're still off a bit though. The Poisson distribution, which Shackleford used in his example, is less precise than the binomial distribution, which I used. Both get "close enough" in this case. But if the binomial distribution is easily calculable, there is little need for the poisson distribution. I'm not sure why Mike used it in some of his answers.

http://www.oxfordmathcenter.com/drupal7/node/297

[alpax]

I agree that it is much more precise to use the binomial distribution than Poisson, should have been the first thing (or basic instinct) when it comes to probability. Mathematical constants like Pi and Euler constant have too many digits to throw off the results, at least Pi has well over 200 million digits after the decimal.

1 in ~11500 is so rare in actuality to a point where (no offense) the casino hit an unlikely jackpot with a player unfortunately, I am probably nearing 3 cycles (15000 hands) myself. I hit them in practice, but never at a live casino. Deuces Wild is pretty brutal with the lack of Deuces as it is accounts about 4 percent of the overall return.

[FloridaPhil]

That's a ton of hands without a quad deuce!  My wife and I played three solid days last trip to Biloxi with only two between us and it wasn't pretty. We have had many multiple quad deuce days, so I guess it averages out.  For us, it's about one every 5,000 hands or so. Here's one I hit yesterday playing the Cheap Strategy at fifty cents.  Got one deuce and drew three more.  Sometimes the stars line up right... 

[alpax]

WTG Phil! The stars always seem to align for you at max bet wagers.

[FloridaPhil]

I hit a lot of single coin quad deuces too, but I don't post pictures of them.  I especially don't post single coin royals!  Playing video poker with the Cheap Strategy takes patience and perseverance, but it works for me.  As my bankroll continues to grow, I am moving up in denomination.  I have enough to play at the 50 cent level now which means I can move to dollar play that much faster.  It's all good...