Question

A professor has two light bulbs in his garage. When both are burned out,they are replaced, and the next day starts with two working light bulbs. Supposethat when both are working, one of the two will go out with probability 0.02 (eachhas probability 0.01 and we ignore the possibility of losing two on the same day).However, when only one is there, it will burn out with probability 0.05. (i) What isthe long-run fraction of time that there is exactly one bulb working? (ii) What is theexpected time between light bulb replacements?

Answer 1

Answer:

(i) 2/7 or 0.2857 of the time

(ii) 70 days

Step-by-step explanation:

(i) There are two possible conditions, two bulbs working or one bulb working. The long run fraction of time that there is exactly one bulb working is given by   the expected time with one light bulb working divided by the expected time with one or two bulbs working:

$F = \frac{E(X=1)}{E(X=1)+E(X=2)}\\ F=\frac{0.05^{-1}}{0.05^{-1}+0.02^{-1}} \\F=\frac{2}{7}=0.2857$

The long-run fraction of time that there is exactly one bulb working is 2/7 or 0.2857 of the time.

(ii) The expected time between light bulb replacements is the expected time for both bulbs to go out:

$T=E(X=1)+E(X=2) = 0.02^{-1}+0.05^{-1}\\T= 70\ days$

The expected time between replacements is 70 days.