In the year 7095, multiple Terran crews are travelling in the Milky Way with an un

stable model of warp drive. If a failure occurs on a warp drive, the

crew has to deactivate it. Each failure is random and independent.

(a) Suppose that the fifirst Terran crew resets their warp drive before every departure. The

amount of time that a failure occurs on the drive can be modeled by an exponential

random variable, where on average it occurs after 20 years. Given that the crew has

been travelling for 30 years with no failure of the drive, what is the probability that

no failure will occur for another 5 years?

(b) Following (a), what is the probability that a failure will occur in the next 10 years?

(c) Suppose that on average 2 warp drives of difffferent crews report a failure each year.

Let X be the number of failures reported within the next year. What type of random

variable is X? Provide your reasoning.

(d) Following (c), use the Poisson approximation to fifind the probability that all crews will

report at least 3 cases within the next year.