Final answer:
To find the probability that out of 4 bolts chosen at random, (i) 1 bolt will be defective, use the binomial distribution formula. The probability is 0.4096. (ii) To find the probability that 0 bolts will be defective, use the same formula. The probability is 0.4096.
Step-by-step explanation:
To find the probability that out of 4 bolts chosen at random, (i) 1 bolt will be defective, we can use the binomial distribution formula. The formula for the probability of exactly k successes in n trials is P(X=k) = (nCk) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and nCk represents the combination of n items taken k at a time. In this case, n = 4, k = 1, and p = 0.2 since 20% of bolts are defective. Plugging these values into the formula, we get P(X=1) = (4C1) * (0.2)^1 * (0.8)^(4-1) = 4 * 0.2 * 0.512 = 0.4096.
(ii) To find the probability that 0 bolts will be defective, we can use the same formula with k = 0. Plugging in the values, we get P(X=0) = (4C0) * (0.2)^0 * (0.8)^(4-0) = 1 * 1 * 0.4096 = 0.4096.