Suppose X, the lifetime of a certain light bulb, is normally distributed with mean 160 hours and standard deviation 30 hours.
a. A randomly selected bulb is observed to be still working after 135 hours. Find the conditional probability that the bulb will blow in the next 60 hours.
b. Let Y be the number of bulbs that are still working after 180 hours among 50 of these bulbs selected independently. Find the exact probability of P(Y > 5).
c. Find P(Y > 5) using Poisson approximation.
d. Find P(Y > 5) using Normal approximation.