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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:

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

(by the craps principle) and

\begin{displaymath}P(win)=P(7\; or\;
11)+(win\vert 4)P(4)+P(win\vert 5)P(5)+P(w...
+ \frac{1}{3}\times\frac{3}{36}

Finally we have enough parts to compute:

\begin{displaymath}P(4\vert win)=\frac{P(win\vert 4)P(4)}{P(win)}=\frac{\frac{1}{3}\times\frac{1}{12}}

Susan Holmes