probability of 6,7 or 8 coming on the flop

[Noo Yawk]

One of the regulars in the room I play made even money bets that a 6, 7 or 8 would hit the flop. He took a few hundred from some players last night doing this. What is his edge here, and how would I do the math?
Thanks in advance.

[BruceZ]

Assuming nobody looked at their cards first, the probability of a 6,7, or 8 is 1 minus the probability of no 6,7,8 = 1 - (40/52)(39/51)(38/50) = 55.3%. He had a 9.4% edge. It would be 14% if he looked at his cards and didn't have one assuming others would always bet if even if they didn't have one.

I've been getting people to give me 2-1 that a pair will appear in 5 cards. It's really about even money, so half the time I win the big bet, and half the time I lose the small bet, so I average winning the small bet. That's a 100% return on my money. One guy asked me if I had no shame, which of course I don't. "It's immoral to let a sucker keep his money." People who can't figure probabilities and bet anyway deserve to lose money to those who can. /forums/images/icons/grin.gif

[Noo Yawk]

Thanks Bruce, I was hoping to get your response.
Yeah, it was quite the display. From what he told me, there were several takers.
Thanks again-NY

[BruceZ]

Here's something interesting about my bet. I have a 100% edge since I win 100% of the money I bet. The suckers only have a 50% disadvantage because they lose 50% of the money they bet. The edge of one player doesn't have to equal the disadvantage of the other player when the amounts wagered are different. In terms of all the money bet by the suckers and me combined, I win 67% of that money, and they win 33%.

[RiverMel]

Thanks, bruce, for giving me two new tricks to try out with friends. I mean, "friends."

[BruceZ]

2-1 on even money is not a 100% return on investment, it's a 50% return on investment. .5(2) - .5(1) = .5. To be precise, this bet is .493(2) - .507(1) = 47.9% edge. The suckers are at a disadvantage of .47.9%/2 = 24% since they lose what I win, but they bet twice as much. Also, I don't win 67% of the money that goes in, only half, but 2/3 of this is my opponent's money.

[Funksolebro]

So what are the odds of any single card coming up on the flop?

[Tekari]

Same thing, 1 - probability of no 6 = 1 - [(48/52)(47/51)(46/50)] = 18.6%