Question

An actuary researches tire failure and determines the probability of an individual tire failing is 0.0025. Assume independence of events for tire failures. a. How many tires will fail on average out of 1000? b. How many tires will fail on average out of 500? c. If 1000 tires are sold, what is the probability that none fail? d. If 1000 tires are sold, what is the probability that at least one will fail?

Answer 1

X ~ Bin ( n , p)

a)

E(X) = n * p = 1000 * 0.0025 = 2.5 3

b)

E(X) = n * p = 500 * 0.0025 = 1.25 1

c)

Since n is large and p is very small, we use poisson approximation,

P(X) = e- * X / X!

Where = n p = 1000 * 0.0025 = 2.5

P(X = 0) = e-2.5

= 0.0821

d)

P(X >= 1) = 1 - P(X = 0)

= 1 - 0.0821

= 0.9179