## Probability and outs

Today JCarver posed a question in IRC that he was asked by someone but wasn’t sure how to answer. It was “If you have 15 outs on the flop, you’re about 30% to hit on the turn, but only 54% to hit by the river, why?” (these numbers are slightly off, but for the statement of the problem it’s fine).

Using “common sense” it may appear to be wrong. That is, if you are 30% with one card to come on the turn, why isn’t it 60% with 2 cards to come. Simply put, probability doesn’t work like this. This situation is tricky in the same way that the three men at a hotel problem is because it’s close to the result you expect.

One easy way to explain it is with different numbers but the same concept. Consider that if you flip a coin once it has a 0.5 chance of being heads. However, if you flip it twice, you can’t add 0.5 and 0.5 and get a 1 probability of it coming up heads. It’s actually 0.75 — you can get HH, HT, TH and TT outcomes, and 3/4 of those have a head in them. This illustrates quite simply why you can’t add two distinct event probabilities.

Furthermore, say you didn’t hit your out on the turn card. The probability of hitting it on the river then goes slightly up because there is now one fewer non-out card in the deck.

For what it’s worth, despite being the math nerd of the Bad Beats Crew, I absolutely hate probability. Common sense will get you in a lot of trouble. That said, there are a lot of interesting paradoxes in probability and statistics such as Simpson’s paradox, but I find game theory strategy to be more satisfying.

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