# Problem 82 Appendix

For those of you coming from The Wizard of Odds this is the derivation of the house advantage on the don't pass in craps.

First review the probability of throwing any given number on any given throw:

• 2: 1/36
• 3: 2/36
• 4: 3/36
• 5: 4/36
• 6: 5/36
• 7: 6/36
• 8: 5/36
• 9: 4/36
• 10: 3/36
• 11: 2/36
• 12: 1/36
If you don't understand why the above is true please review my page of dice probabilities basics.

From the above the probalility of rolling 2 is 1/36 and the probability of rolling an 3 is 2/36 = 1/18. These are the numbers that win on the first throw.

Next lets assume the point thrown on the first roll is a 4, what is the probability of throwing a 7 before another 4?

Let pr(x) stand for the probability of event x happening on any given roll. The answer is:

pr(7) +
pr(anything other than 4 and 7) * pr(7) +
pr(anything other than 4 and 7)2 * pr(7) +
pr(anything other than 4 and 7)3 * pr(7) +
pr(anything other than 4 and 7)4 * pr(7) +
+ ...

Pr(4) = 6/36 = 1/6, pr(anything other than 4 and 7) = 1-3/36-6/36 = 27/36 = 3/4.

pr(rolling a 4 before a 7)
= 1/6 + (3/4 * 1/6) + ((3/4)2 * 1/6) + ((3/4)3 * 1/6) + ...
= 1/6 * sum for i = 0 to infinity of (3/4)i
= 1/6 * (1/(1-3/4))
= 1/6 * 4 = 2/3.

The probability of rolling a 10 is the same as the probability of rolling a 4 so pr(rolling a 7 before a 10) also equals 2/3.

Next assume the point thrown is a 5.

Pr(5) = 4/36 = 1/9, pr(anything other than 5 and 7) = 1-4/36-6/36 = 26/36 = 13/18.

pr(rolling a 7 before a 5)
= 1/6 + (13/18 * 1/6) + ((13/18)2 * 1/6) + ((13/18)3 * 1/6) + ...
= 1/6 * sum for i = 0 to infinity of (13/18)i
= 1/6 * (1/(1-13/18))
= 1/6 * 18/5 = 3/5.

The probability of rolling a 9 is the same as the probability of rolling a 5 so pr(rolling a 7 before a 9) also equals 3/5.

Next assume the point thrown is a 6.

Pr(6) = 5/36, pr(anything other than 6 and 7) = 1-5/36-6/36 = 25/36.

pr(rolling a 7 before a 6)
= 1/6 + (25/36 * 1/6) + ((25/36)2 * 1/6) + ((25/36)3 * 1/6) + ...
= 1/6 * sum for i = 0 to infinity of (25/36)i
= 1/6 * (1/(1-25/36))
= 1/6 * 36/11 = 6/11.

The probability of rolling an 8 is the same as the probability of rolling a 6 so pr(rolling a 7 before a 8) also equals 6/11.

The probability of winning the don't pass bet is:
pr(2) + pr(3) + pr(4)*pr(7 before a 4) + pr(5)*pr(7 before a 5) + pr(6)*pr(7 before a 6) + pr(8)*pr(7 before a 8) + pr(9)*pr(7 before a 9) + pr(10)*pr(7 before a 10)
= 1/36 + 1/18 + (1/12 * 2/3) + (1/9 * 3/5) + (5/36 * 6/11) + (5/36 * 6/11) + (1/9 * 3/5) + (1/12 * 2/3)
= 1/36 + 1/18 + 1/18 + 3/45 + 30/396 + 30/396 + 3/45 + 2/36
= 55/1980 + 110/1980 + 110/1980 + 132/1980 + 150/1980 + 150/1980 + 132/1980 + 110/1980
= 949/1980.

The probability of a tie by throwing a 12 on the first roll is 1/36. The probability of throwing anything other than a 12 is 35/36. It is common practice to ignore ties when calculating expected returns and house advantages when applied to casino gambling. So the probability of winning, given that there was no tie is (949/1980) / (35/36) = (949*36)/(1980*35) = 949/1925 =~ 49.3% .

The expected return on this wager is the product of the probability of winning and the ratio of what you keep (including your original wager) to the original bet if you do win (in this case 2). Thus the expected return is 949/1925 * 2 = 1898/1925 =~ 98.6%.

The house advantage is what the casino gets to keep, on average, which is 1 minus the expected return, which equals 1-(1898/1925) = 27/1925 =~ 1.4026%.

Note that if we did not ignore ties the expected return would be (2*949)/1980 + 1/36 = 0.9864. The house advantage would be 1 - 0.9864 = .013636, or 1.3636%.

Michael Shackleford, A.S.A., March 10, 1999