Question

3. The U.S. Postal Service claims that only 85% of its Express Mail is actually delivered company sends out 20 packages to clients using Express Mail. Assume that the package arrivals are independent and assume that the Postal Service's claim is true (a) What is the probability that at least 1 of the packages is not delivered on-timei (b) What is the probability that exactly 18 of the packages are delivered on-time? (c) What is the probability that fewer than 18 of the packages are delivered on-time?

Answer 1

Given that, P = 0.85 and n = 20 let X ~ Binomial (n = 20, P = 0.85) P(X = x) = n_c_x P^x (1 - P)^n - x: x = 0, 1, - - - - n a) The probability that at least 1 of the package is not delivered on - time ? P(X greaterthanorequalto 1) = 1 - P(X < 1) = 1 - P(X = 0) = 1 - {20_c_0 (0.85)^0 (1 - 0.85)^20 - 0} = 1 - {(1) (1) (0.15)^20} = 1 - (0.15)^20 = 1 - 0 P(X greaterthanorequalto 1) = 1 \r\n b) The probability that exactly 18 of the packages are delivered on - time is P(X = 18) = 20_c_ 18 (0.85)^18 (1 - 0.85)^20 - 18 = 20!/18! (20 - 18)! (0.85)^18 (0.15)^2 = (190) (0.85)^18 (0.15)^2 P(X = 18) = 0.2293 c) The probability that Fewer than 18 of the packages are delivered on - time