Follow up to Jason and Marshall's prop bet last night. Jason offers Marshall his choice of taking paired flop (which presumably includes three of a kind) versus an all low flop (all three cards a different rank of 8 or lower). I consider myself not a slouch at math so I can't understand why I'm having such a tough time at this one.

At the time I thought that paired board would have better odds. Just seemed that paired flops would be easier than getting all three cards to do the same thing. Then the next hand came along and I didn't think about it any more until I got home. I tried to crunch numbers to see which one had the edge but couldn't get numbers that sounded right.

For low, I figured it is a 32/52 chance of the first card being low (four each of A-8), second card has only seven ranks left so a 28/51 chance for that, and finally 24/50 for the third card to be low as well. So (32/52) * (28/51) * (24/50) = .615 * .549 * .480 = .162 or roughly 16%. Back of the envelope calculations say that three low cards are roughly three coin flips in a row which would be 1/8 or 12.5% so that seems correct. However I know that for some calculations you actually have to find the odds of *missing* then multiply those together. Since all of these values are close-ish to 50/50 I could be doing the wrong method but still getting a similar number.

Then for the paired board I figured the odds of the first card coming from the deck are 100%. Easy peasy. Now the next card matching it would be 3/51 or .058 and the odds of the third card matching either of the two others would be 6/50 or .120. But multiplying those numbers together gives you .007 or less than 1% and that seemed way wrong. Actually as I was blogging I think that I need to ADD them together since they are independent events. Is that right? So .058 + .120 = .178 or almost 18%. Do paired boards really show up almost 1/5 of the time? Back of the envelope says that three cards need to make a pair at least as often as two cards do so the correct number must be > 6% but is it that much more?

Are those the right numbers? Paired flop 17.8% favored over all low flop 16.2%?

## Wednesday, February 27, 2008

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## 6 comments:

Your low calculations are spot on, and your logic on the paired board is

almostright; that's actually the case where you want the 'miss' odds (i.e., what are the odds all three are different) and to invert those. Basically, if A and B are independent events, and the probability of A happening is x and the probability of B happening is y, then the probability of A AND B happening is x*y, but the probability of A OR B happening is the inverse of the probability that neither happens -- or 1-(1-x)*(1-y). In this case, you're spot on for your first calculation; for the second, the best thing to do is to essentially do the equivalent of the 'OR' calculation, and turn the problem on its head. The probability that the second card is different than the first is 48/51; the probability that the third card is different than both of the first two (contingent on the second being different than the first) is 44/50. Multiplying these two together gets you an 82.8% chance of all three cards being distinct, or a 17.2% chance of the board being paired; a bit better than one in six, which seems about right to me. Notably, if you remove the 'all three different' restriction from the all-low flop, then the odds go to (32/52)*(31/51)*(30/50), or about 22% of the time that your 9s will be an overpair...very helpful guys, I'll come up with some more prop bets in the future for your enjoyment.

I used to be able to easily pull this kind of math off but now am reliant on the young whiz kid generation. Kind of cool how the 2 events came out to almost the same probability.

Thanks for the math and the explanation Steve. Now I'll be ready for the next prop bet proposal.

Damn you guys are awesome. I feel slightly dumber than I did before reading that, but I will say I at least understand it.

Nice post, and interesting how close it really is.

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