Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the other lifetimes, and that each lifetime has an exponential distribution with parameter 3.
(a) What is the probability that all ten bulbs fail before time t?
(b) What is the probability that exactly k of the ten bulbs fail before time t?
(c) Suppose that nine of the bulbs have lifetimes that are exponentially distributed with parameter λ
and that the remaining bulb has a lifetime that is exponentially distributed with parameter θ (it is made by another manufacturer). What is the probability that exactly five of the ten bulbs fail before time t?
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