A box contains 20 lightbulbs, of which 5 are defective.

If 4 lightbulbs are picked from the box randomly, the probability that at most 2 of them are defective is

a. 31/969

b. 70/323

c. 91/323

d. 938/969

The given question "A box contains 20 lightbulbs, of which 5 are defective. If 4 lightbulbs are picked from the box randomly, the probability that at most 2 of them are defective is" is a binomial distribution probability question. The probability of an event x, occuring in a binomia distribution is given by: P (x) = nCx (p) ^x (q) ^ (n - x), where: n is the number of trials = 4, x is the event of interest = at most 2, p is the probability of success = probability that a lightbulb is defective = 5/20 = 1/4, q is the probability of failure = probability that a lightbulb is not defective = 1 - 1/4 = 3/4. P (at most 2) = P (0 or 1 or 2) = P (0) + P (1) + P (2) = 4C0 (1/4) ^0 (3/4) ^ (4 - 0) + 4C1 (1/4) ^1 (3/4) ^ (4 - 1) + 4C2 (1/4) ^2 (3/4) ^ (4 - 2) = 1 x 1 x (3/4) ^4 + 4 x 1/4 x (3/4) ^3 + 6 x 1/16 x (3/4) ^2 = 81/256 + 27/64 + 27/128 = 243/256