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%?