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## Homework Statement

A traffic signal operates in the following cyclic regime: amber (A) light for 5 seconds,

then red (R) for 30 seconds, then amber again for 5 seconds, then green (G) for 40 seconds

(thus making a cycle ARAG), and then in the cyclic manner, i.e. ARAGARAG... .

Let us assume that the amber, green and red bulbs can fail every time they switch on

with independent probabilities p

_{A}, p

_{G}and p

_{R}, respectively.

What is the mean number of cycles of non-failure operation of the red bulb?

## Homework Equations

I suppose the E(X)=Σ(x*P(X=x)) over all valid x is probably relevant here.

## The Attempt at a Solution

The probability that the bulb operates for n cycles without failure is P(No Fail)^n = (1-p

_{R})

^{n}. So then E(N)=Σ(n*(1-p

_{R})

^{n}) where n is summed from 0 to infinity perhaps? But I wouldn't be able to reduce this (and from the solution, it isn't right anyway ...)