Question

The weight X (in pounds) of a roll of steel is a random variable with probability density function given by f(x) = (x - 499)/2 if 499 < x < 501 and zero otherwise. A roll is considered defective if its weight is less than 500 pounds.
(a) If 3 independent rolls of steel are examined, find the probability that exactly one is defective.
(b) If 64 independent rolls of steel are examined, approximate the probability that at least 20 of them are defective.

Thank you!

Answer 1

a) The probability that a roll is defective is computed here as:

P(X < 500)

This is computed as:

Therefore 0.25 is the required probability here.

Given that 3 independent rolls of steel are examined, the number of those rolls which are defective is modelled here as:

Probability that exactly one is defective is computed here as:

Therefore 0.421875 is the required probability here.

b) For 64 independent rolls of steel are examined, the probability that at least 20 are defective is computed as:

P(X >= 20) = 1 - P(X <= 19)

This is computed in EXCEL as:
=1-binom.dist(19,64,0.25,TRUE)

0.1561 is the output here.

Therefore 0.1561 is the required probability here.