Answer: The Craps Shooter's Odds

Key Idea / Intuition

The game splits into independent cases based on the first roll. For the "point" scenarios, only two outcomes matter: rolling the point again vs. rolling a 7. Once the point is established, all other outcomes are irrelevant — they just delay the resolution. So the conditional win probability given a point $p$ is simply the probability of rolling $p$ divided by the probability of rolling $p$ or $7$. Then we weight by the probability of each point occurring.

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Formal Proof / Solution

Step 1: First-Roll Win/Loss

Count ways to roll each total with two dice (out of 36 equally likely outcomes):

| Total | Ways | Probability | |-------|------|-------------| | 2 | 1 | 1/36 | | 3 | 2 | 2/36 | | 7 | 6 | 6/36 | | 11 | 2 | 2/36 | | 12 | 1 | 1/36 |

Immediate win (7 or 11): $\frac{6+2}{36} = \frac{8}{36}$

Immediate loss (2, 3, or 12): $\frac{1+2+1}{36} = \frac{4}{36}$

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Step 2: Point Probabilities and Conditional Win

For a point $p$, all other rolls are irrelevant. By the geometric trials argument, the conditional probability of winning given point $p$ is:

$P(\text{win} \mid \text{point is } p) = \frac{P(\text{roll } p)}{P(\text{roll } p) + P(\text{roll } 7)}$

Compute for each possible point:

| Point $p$ | Ways to roll $p$ | $P(p)$ | $P(\text{win}\mid p) = \frac{P(p)}{P(p)+6/36}$ | |-----------|-----------------|---------|----------------------------------------------| | 4 | 3 | 3/36 | $\frac{3}{3+6} = \frac{1}{3}$ | | 5 | 4 | 4/36 | $\frac{4}{4+6} = \frac{2}{5}$ | | 6 | 5 | 5/36 | $\frac{5}{5+6} = \frac{5}{11}$ | | 8 | 5 | 5/36 | $\frac{5}{5+6} = \frac{5}{11}$ | | 9 | 4 | 4/36 | $\frac{4}{4+6} = \frac{2}{5}$ | | 10 | 3 | 3/36 | $\frac{3}{3+6} = \frac{1}{3}$ |

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Step 3: Total Win Probability

$P(\text{win}) = \frac{8}{36} + \sum_{\text{points}} P(\text{first roll} = p) \cdot P(\text{win}\mid p)$

$= \frac{8}{36} + 2\left[\frac{3}{36}\cdot\frac{1}{3} + \frac{4}{36}\cdot\frac{2}{5} + \frac{5}{36}\cdot\frac{5}{11}\right]$

Compute each term:

$\frac{3}{36}\cdot\frac{1}{3} = \frac{3}{108} = \frac{1}{36}$

$\frac{4}{36}\cdot\frac{2}{5} = \frac{8}{180} = \frac{2}{45}$

$\frac{5}{36}\cdot\frac{5}{11} = \frac{25}{396}$

Multiply each by 2 (for symmetric pairs 4&10, 5&9, 6&8):

$2\left[\frac{1}{36} + \frac{2}{45} + \frac{25}{396}\right]$

Find common denominator (LCM of 36, 45, 396 = 1980):

$= 2\left[\frac{55}{1980} + \frac{88}{1980} + \frac{125}{1980}\right] = 2\cdot\frac{268}{1980} = \frac{536}{1980} = \frac{134}{495}$

Total:

$P(\text{win}) = \frac{8}{36} + \frac{134}{495} = \frac{440}{1980} + \frac{536}{1980} = \frac{976}{1980} = \boxed{\frac{244}{495} \approx 0.4929}$

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The Surprise

The house edge is only about 1.41% — craps is one of the fairest casino games in existence, with the player winning just barely under half the time. This elegant near-symmetry is why craps became so popular.