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ballparkfrank
Posted by ballparkfrank
Nov 22, 2013

Odds of getting 4 of a kind in video poker after holding 3 of a kind

Thanks in advance. I must be missing something; not always the sharpest knife in the drawer.

On the W of Oz website, it says that after holding 3 of a kind in video poker, the odds of getting 4 of a kind are 1 in 23.5.

It seems to me that if you hold 3 of a kind, and there are 48 cards left, which contain only one of whatever card you held (A for example) the prob of getting 4 of a kind would be 1 in 48. What am I missing???

Comments

JB
JB Nov 22, 2013

First of all, there are not 48 cards left, there are 47 (the full 52 minus the 5 you were dealt).

Second, if you're holding 3 of a kind, you get 2 replacement cards, not just one. 2 in 47 = 1 in 23.5



That's just a simplification. Here is the bigger picture:



There are 47 * 46 / 2 = 1081 combinations of possible replacement cards. (The first replacement card can be any of the 47 cards left in the deck, then after that has been dealt, the other replacement card can be any of the 46 cards left in the deck. And since the order in which the two replacement cards are dealt does not matter, we divide by 2! [the factorial of 2] since 2 cards were dealt.)



Then, we know that to get 4 of a kind we need one specific card, and the other card can be any of the other 46 cards. So 1*46 = 46 combinations will result in a 4 of a kind, and 46/1081 is the same as 2/47 or 1 in 23.5.