Problem 89 Solution

Solution 1

The house advantage in just the pass line bet is 7/495, see the solution to problem 82 for an explanation of this step.

There is a 1/3 chance that the come out roll will be a 2,3,7,11, or 12, click here if this step needs explanation.

There is a 2/3 chance that double odds will be taken. Given that $1 is bet on the pass line the expected money bet per round is 1/3*$1 + 2/3*$3 = $7/3. Since $1 is always bet on the pass line 1/(7/3)=3/7 of money bet, overall, is on the pass line, and the other 4/7 is on the odds.

Thus the house advantage is 3/7*(7/495) + 4/7*0 = 3/495 = 1/165. The players advantage is just the opposite of this, -1/165.

Solution 2

The player's advantage is the sum over all possible outcomes of the product of the probability of the outcome and the net profit or loss, divided by the sum over all possible outcomes of the expected amount bet.

Lets say the original wager on the pass line is 5. By summing the following possible outcomes you will have the expected gain or loss.

• Roll 2 on come out roll: 1/36 * -5 = -5/36 = -4950/35640
• Roll 3 on come out roll: 2/36 * -5 = -10/36 = -9900/35640
• Roll 4 on come out roll, then win: 3/36 * 3/9 * (5+20) = 225/324 = 24750/35640
• Roll 4 on come out roll, then lose: 3/36 * 6/9 * (-5-10) = 270/324 = -29700/35640
• Roll 5 on come out roll, then win: 4/36 * 4/10 * (5+15) = 320/360 = 31680/35640
• Roll 5 on come out roll, then lose: 4/36 * 6/10 * (-5-10) = -360/360 = -35640/35640
• Roll 6 on come out roll, then win: 5/36 * 5/11 * (5+12) = 425/396 = 38250/35640
• Roll 6 on come out roll, then lose: 5/36 * 6/11 * (-5-10) = -450/396 = -40500/35640
• Roll 7 on come out roll: 6/36 * -5 = -30/36 = 29700/35640
• Roll 8 on come out roll, then win: 5/36 * 5/11 * (5+12) = 425/396 = 38250/35640
• Roll 8 on come out roll, then lose: 5/36 * 6/11 * (-5-10) = -450/396 = -40500/35640
• Roll 9 on come out roll, then win: 4/36 * 4/10 * (5+15) = 320/360 = 31680/35640
• Roll 9 on come out roll, then lose: 4/36 * 6/10 * (-5-10) = -360/360 = -35640/35640
• Roll 10 on come out roll, then win: 3/36 * 3/9 * (5+20) = 225/324 = 24750/35640
• Roll 10 on come out roll, then lose: 3/36 * 6/9 * (-5-10) = 270/324 = -29700/35640
• Roll 11 on come out roll: 2/36 * 5 = 10/36 = 5/18 = 9900/35640
• Roll 12 on come out roll: 1/36 * -5 = -5/36 = -4950/35640

Now lets sum the expected amount wagered:

If the first roll is a 2,3,7,11, or 12 the wager is 5.
If the first roll is anything else the wager is 15.
Thus the expected wager is (1+2+6+2+1)/36 * 5 + (3+4+5+5+4+3)/36 * 15 = 60/36 + 360/36 = 420/36 = 35/3.

Thus the player's advantage is (-2520/35640) / (35/3) = -7560/1,247,400 = -378/62370 = -1/165.

In other words the house has an advantage of approximately 0.61%.