Twist on the old 6-7-8 prop bet

[LondonBroil]

So the players in my home game are now wise to this. But what if I made some adjustments to it? If I adjusted it to..

6, 7, or 8 on the flop: I win the bet

66x, 77x, or 88x on the flop: I pay them double the bet

666, 777, or 888 flops: I pay them triple


Do I still have the advantage? If so, how much more of an advantage would I lose if I added 67x and 78x to the 2nd criteria?

[BruceZ]

[ QUOTE ]
So the players in my home game are now wise to this. But what if I made some adjustments to it? If I adjusted it to..

6, 7, or 8 on the flop: I win the bet

66x, 77x, or 88x on the flop: I pay them double the bet

666, 777, or 888 flops: I pay them triple


Do I still have the advantage?

[/ QUOTE ]

No, your opponents would have an edge of 1.36% even though you still win more hands.

P(no 6,7,8) = C(40,3) / C(52,3) = 44.71%

P(66x or 77x or 88x w/no trips) = 3*6*48 / C(52,3) = 3.91%

P(666 or 777 or 888) = 3*4 / C(52,3) = 0.05%

P(single 6,7, or 8) = 1 - P(no 6,7,8) - P(66 or 77 or 88) - P(666 or 777 or 888) = 51.33%

edge = -1*44.71% -2*3.91% -3*0.05% + 1*51.33% = -1.36%

What's happening here is that the hands you are giving up occur about 4% of the time, and even though they represent a small percentage of the hands you used to win, letting your opponent win these hands instead of you would represent a swing of about 8% if he only won 1 bet. This is almost as large as the edge you had before of about 10%, and since he now wins an additional bet, that is enough to reverse the edge. You could maintain an advantage of 2.55% by only giving him 1 bet on the pairs, even if you still gave him 3 bets on the trips.

[ QUOTE ]
If so, how much more of an advantage would I lose if I added 67x and 78x to the 2nd criteria?

[/ QUOTE ]

Your opponents would have an edge of 19.6%. Now you are reversing hands that occur even more often. Since this may be counterintuitive to your opponents, perhaps you could take the other side of this bet. /images/graemlins/cool.gif

P(67x or 78x w/no pairs) = (4*4*44 + 4*4*40) / C(52,3) = 6.08%

P(single 6,7 or 8 w/no 67 or 78) = 1 - P(no 6,7,8) - P(66 or 77 or 88) - P(666 or 777 or 888) - P(67 or 78) = 45.25%

The other probabilities are the same as before (including 667, 677, 778, and 788).

edge = -1*44.71% -2*3.26% -3*0.05% -2*6.08% + 1*45.25% = -19.6%

[LondonBroil]

Thanks for the math. I'll try to take the other side of this bet next time. /images/graemlins/wink.gif