Calculating pot odds.

[Crooked Paul]

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Why is it that burn cards and other players hole cards are not taken into account? They can and DO make a difference, is it simply because they are unknown?

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Thought I'd give you a more complete answer to this question, because among my poker-playing friends a lot of them didn't grasp this at first.

When you figure drawing odds, you're using the information you have (all the cards you have seen) to make inferences about the information you don't have (all the cards you haven't seen). You can do this because you have reliable information about the deck as a whole. So if there's an ace of spades on the board, you know for a fact that it isn't among the unseen cards, whether in someone's hand, the burn pile, or the rest of the deck.

Pretty obvious. So far so good. The important thing to realize is that the conclusions you draw from the cards you've seen always apply to all the unseen cards. People seem to have trouble keeping this in mind because the next board card is all they care about.

So when you say your chance to complete your flush on the turn is 9/47, that's true, but only because the chance that any unseen card you pick (from someone else's hand, from the burn pile, from the bottom of the deck, the top of the deck) has that same 9/47 chance to match the suit of your four flush.

People seem to get confused because the unseen cards are in multiple different categories (muck, opponents' hands, undealt deck). But that doesn't matter at all. Here's an equivalent example that illustrates the point:

If you have 20 total marbles in a bag, 15 white and 5 black, and you choose one marble from the bag at random, the chance that it will be black is 5/20 = 1/4. Now suppose you take 2 marbles at random without looking at them and put them in a box, and you take 7 marbles at random without looking at them and flush them down the toilet.

There are 11 marbles left in the bag. According to the information you have, if you choose one at random what is the chance it will be black?

....if you said anything other than 1/4 then this post has failed utterly. =)



Crooked

[wdeadwyler]

I see what you are saying. The expected ratio of black marbles to white marbles does not change because each time we remove a marble, the overall probability is still 1/4, even if n has now decreased.

[Crooked Paul]

Right. The key thing to realize is that separating the unseen cards/marbles into different groups doesn't give us any new information, so the old information applies equally to them all, no matter where they are.

Before the turn is dealt, you could cut the deck and take, say, 25 cards and put them right into the burn pile without looking at them, and the chance that the next card on top of the live deck will match your flush is still 9/47, even though there are less than 22 cards left in the deck at that point. (I say "less than 22" because of the two unseen cards in each player's hand, which varies.) No matter how many players are in the hand or how many cards you burned on a whim, that doesn't change that there are still 47 total unseen cards, and your inferences apply equally to them all, including the only one you care about, which is the next board card.


Crooked