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 pA, pG and pR, respectively.
What is the mean number of cycles of non-failure operation of the red bulb?
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-pR)n. So then E(N)=Σ(n*(1-pR)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 ...)