# Multiple die rolls: You roll a fair die until you see a five, then a six; after that, you stop….

Multiple
die rolls: You roll a fair die until you see a five, then a six; after that,
you stop. Write P.N/ for the probability that you roll the die N times. (a)
What is P(1)?

(b)
Show that P(2) = (1=36).

(c)
Draw a directed graph encoding all the sequences of die rolls that you could
encounter. Dont write the events on the edges; instead, write their
probabilities. There are five ways not to get a five, but only one probability,
so this simplifies the drawing.

(d)
Show that P(3) = (1/36).

(e)
Now use your directed graph to argue that P(N) =(5/6)P(N-1) C (25/36)P(N- 2).

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