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Originally Posted by ScotchOnDaRocks
I'm not sure what you did above
That's too bad. I had hoped to make my reasoning clear.
After the flop, how many cards are you looking at? (The answer is 8... the three from the flop plus the five in your hand).
So after the flop, how many cards are in the group of unseen cards? The answer is 44 (from 52-3-5=44).
And how many of these 44 cards can be fives? (The answer is 2, since you're looking at one five in your hand and one five on the board).
And how many groups of 4 cards can be made from 44 cards? (The answer is 11, since 44/4=11).
And how many of these 11 groups of 4 cards each can contain a five? (2 is the answer).
- digression: Maybe you're thinking one of the 11 groups of four cards can contain two fives, which is correct, and which slightly decreases the chances two of the eleven groups of four cards contain a five.
Actually there are 135751 possible ways to choose 4 cards from 44 cards.
861 of these contain both missing fives.
22960 contain exactly one five.
111930 contain no fives.
So 23821+861=24682 of these 135751 possible ways to choose 4 cards contain either one five or two fives.
Thus it's actually 111930 to 22960, or 4.875 to 1, close to 5 to 1, that four drawn cards won't contain at least one five.
And, continuing my approximation method, if two of the eleven of these groups of 4 cards contain a five, how many of the eleven of these groups don't contain a five? (9 is the answer)
24682/135751= a probability of 0.181818 that a chosen group of 4 cards will contain at least one five.
And the corresponding probability a chosen group of 4 cards will NOT contain at least one five must be 0.818182
So it's approximately 0.818181 to 0.181818 or 9 to 2 against at least one five in any group of four cards we choose. (divide each of those decimal fractions by 9 to clearly see the 9 to 2 ratio).
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but I think the odds may be slighty worse than that.
You're right. The odds are 4.875 to 1, not the 4.50 to 1 I got with my approximation.
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Basically it's like you are trying to flop a set with pocket fives but you get a four card flop instead of three.
Correct.
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Seems like it would be more like 6 to 1.
I think the odds of Hero catching at least one five are 4.875 to 1. (But there are some other ways Hero can win... for example maybe by catching a deuce).
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And you are only going for half the pot.
Good point.
Buzz
Last edited by Buzz; 11-01-2016 at 01:19 PM.
Reason: correct typo omission