A Craps game consists of throwing 2 dices. If the sum is either 7 or 11, you win. Else, if the sum is either 2,3 or 12, then you loose. If the sum is either 4, 5, 6, 8, 9 or 10, then let's call the sum $X$. You will throw the two dices again, until the sum 7 (you loose) or the sum is $X$ (you win). What is the probability of winning the game?
I made some calculations, yet I want to verify if the result is right.
The probability of winning in the first throw, $V_1$ is:
$$P(V_1) = \frac{\{\text{sum is either 7 or 11}\}}{36} = \frac{8}{36}$$
Assume the sum is $x \in X =\{4,5,6,8,9,10\}$.
The probability of winning in the second throw, knowing the result of your first throw is $x$ is given by:
$$P(V_2) = 2\left(\frac{3^2}{36^2}+\frac{4^2}{36^2} +\frac{5^2}{36^2}\right) $$
Because $$ P(X=4)=P(X=10)=\frac{3}{36} $$
$$ P(X=5) = P(X=9) = \frac{4}{36} $$
$$ P(X=6) = P(X=8) = \frac{5}{36} $$
Following the same track of thought, one may have the probability of winnin in the $\text{$3^{rd}$}$ throw is:
$$ P(V_3) = 2\left( \frac{3^2}{36^2}\frac{27}{36}+\frac{4^2}{36^2}\frac{26}{36} + \frac{5^2}{36^2}\frac{25}{36}\right) $$
Because you need the second throw not to be $7$ neither the sum you had in first round $X$.
We can extend the probability of winning in the n-th round as:
$$ P(V_n) = \frac{3^2}{36^2} \left(\frac{27}{36}\right)^{n-2} +\frac{4^2}{36^2} \left(\frac{26}{36}\right)^{n-2} + \frac{5^2}{36^2}\left(\frac{25}{36}\right)^{n-2}$$
So we can evaluate the sum as a geometric series, when $ 0 < \alpha <1$:
$$ \sum_{n=0}^{\infty} \alpha^n = \frac{1}{1-\alpha} $$
So we have the probability of winning:
$$ P(V) = \frac{8}{36} + 2\left( \frac{3^2}{36^2} \frac{36}{9} + \frac{4^2}{36^2}\frac{36}{10} + \frac{5^2}{36^2}\frac{36}{11} \right) $$
$$ P(V) = \frac{8}{36} + 2\left( \frac{1}{36} + \frac{25}{36\cdot 5} + \frac{25}{18 \cdot 11} \right) \approx 0,493 $$
What do you think? Is it right?
Best Answer
I'd attack the problem of whether you successfully score a point before crapping out with a $7$ a little more simply. If you have a point, the only issue is whether you roll that point before you roll a $7$. All other throws are irrelevant so you don't need to walk through the geometric series. Just use a Bayesian calculation -- given that a throw is either a $z$ or a $7$ (where $z$ is your point), what is the probability that it's a $z$?
So if your point is $4$ or $10$, your chance of winning is $\frac{1}{3}.$
If your point is $5$ or $9$, your chance of winning is $\frac{2}{5}$.
If your point is $6$ or $8$, your chance of winning is $\frac{5}{11}$.
So by my calculation
$$P(V)=\frac{2}{9}+\frac{1}{6}\frac{1}{3}+\frac{2}{9}\frac{2}{5}+\frac{5}{18}\frac{5}{11} = \frac{244}{495} \approx 49.29\%.$$