Example of computation for craps 10/19

Suppose I said I had just won at craps, and I ask you what the probability of my having rolled a 4 in the first place was?

In terms of events I want P(4 first throw | win), in order to compute this we have to know P(win) and $P(win\vert 4 \; at \; first\; throw)$, these are both directly computable, in class I showed that:

$$P(win\vert 4 \; at \; first\; throw)= \frac{3}{3+6}=\frac{1}{3}$$

(by the craps principle) and

$$P(win)=P(7\; or\; 11)+(win\vert 4)P(4)+P(win\vert 5)P(5)+P(w... ...{4}{36} + \frac{1}{3}\times\frac{3}{36} =\frac{244}{495}=.4929$$

Finally we have enough parts to compute:

$$P(4\vert win)=\frac{P(win\vert 4)P(4)}{P(win)}=\frac{\frac{1}{3}\times\frac{1}{12}} {.4929}=.0564$$