Ask The Wizard #187
We can see from my analysis of the Fire Bet that the probability of a shooter making all six points is 0.000162435. So, the value of the promotion per shooter is $4,000 × 0.000162435 = 0.649739.
The next question to be asked is what is the expected loss per shooter. The house edge on the pass line bet is 7/495 = 1.414141%. The tricky part is how many pass line bets will a shooter make, on average.
There are four possible states the shooter can be in. Let's define each one as the expected number of future pass line bets for that shooter.
- A = Come out roll
- B = Point of 4 or 10 made
- C = Point of 5 or 9 made
- D = Point of 6 or 8 made
Here are the equations showing the probability of each state leading to the next state.
A = 1 + (12/36)*A + (6/36)*B + (8/36)*C + (10/36)*D
B = (1/3)*A
C = (2/5)*A
D = (5/11)*A
A little algebra results in A = 2.525510, the number of pass line bets made per shooter.
So, the expected loss per $5 shooter is $5*2.525510*0.0141414 = 0.178571.
The expected amount bet by the shooter is $5*2.525510=$12.627551.
Finally, the expected return is the expected win divided by the expected bet: (0.649739-0.178571)/12.627551 = 3.73127%. So the house edge is -3.73%.
Required Banker Ratio to Break Even in Pai Gow Poker
7/2 wins: 57.07%
6/2 wins: 25.13%
However that doesn’t prove that 7/2 is the superior hand in general. Considering only your own hand, according to my various power rating tables, 7/2 off-suit is worse than 6/2 off-suit. For four or fewer players, the worst hand is 3/2 off-suit.
So, if the bet was phrased as who would win between 6/2 and 7/2 offsuit, then you win. However that doesn’t prove that 7/2 is the better hand in general, only for that one situation.