Final answer:
A student asked questions about probabilities based on past data related to customer calls being resolved within one hour. The questions asked for probabilities of different scenarios based on the likelihood of calls being resolved. The detailed answer provides step-by-step explanations and calculations for each question.
Step-by-step explanation:
(A) To find the probability that no problem would be resolved within one hour, we need to find the probability that all 10 calls are not resolved within one hour. Since the past data indicate that the likelihood is 0.80 that a customer call is resolved within one hour, the probability that a call is not resolved within one hour is 1 - 0.80 = 0.20. Therefore, the probability that no problem would be resolved within one hour is 0.20 to the power of 10, which is approximately 0.00001.
(B) To find the probability that exactly 7 calls will be resolved within one hour, we need to find the probability of getting 7 successes (resolved calls) out of 10 trials (customer calls). We can use the binomial probability formula: C(10, 7) * (0.80)^7 * (0.20)^(10-7) = 120 * 0.3277 * 0.008 = approximately 0.314.
(C) To find the probability that at most 3 calls will be resolved within one hour, we need to find the probabilities of getting 0, 1, 2, and 3 successes out of 10 trials and sum them up. The probability that 0 calls will be resolved is (0.20)^10 = approximately 0.00001. The probability that 1 call will be resolved is C(10, 1) * (0.80)^1 * (0.20)^(10-1) = 10 * 0.80 * 0.1074 = approximately 0.859. The probability that 2 calls will be resolved is C(10, 2) * (0.80)^2 * (0.20)^(10-2) = 45 * 0.64 * 0.028 = approximately 0.725. The probability that 3 calls will be resolved is C(10, 3) * (0.80)^3 * (0.20)^(10-3) = 120 * 0.512 * 0.004 = approximately 0.196. The sum of these probabilities is 0.00001 + 0.859 + 0.725 + 0.196 = approximately 1.780.
(D) To find the probability that at least 4 calls will be resolved within one hour, we can find the probability of getting 0, 1, 2, 3 successes and subtract it from 1. The probability that 0, 1, 2, and 3 calls will be resolved is approximately 1.780 (found in part C). Therefore, the probability that at least 4 calls will be resolved is 1 - 1.780 = approximately -0.780. Since a probability cannot be negative, the probability of this event is 0.
(E) To find the probability that between 1 to 8 calls (inclusive) will be resolved within one hour, we need to find the probabilities of getting 1, 2, 3, 4, 5, 6, 7, and 8 successes out of 10 trials and sum them up. The probabilities for each number of successes can be found using the binomial probability formula. Adding up the probabilities for 1 to 8 successes gives us approximately 0.404.
(F) The average number of customers who would have their service problems resolved within one hour can be found using the expected value formula. The expected value is the probability of success (0.80) multiplied by the number of trials (10). Therefore, the expected number of customers with resolved problems within one hour is 0.80 * 10 = 8.