Question

Suppose that one of the students takes the examination answers each of the questions with an independent random guess. Let Y be the number of questions answered correctly. If Y has a binomial distribution, compute P(Y<7). Compute P(1<Y<5).

Answer 1

n = 15

p = 1/4 = 0.25

P(Y= y) = 15Cy * 0.25^y * (1 - 0.25)^15-y

a) P(Y < 7) = P(Y = 0) + P(Y = 1) + P(Y = 2) + P(Y = 3) + P(Y = 4) + P(Y = 5) + P(Y = 6)

                  = 15C0 * 0.25⁰ * 0.75¹⁵ + 15C1 * 0.25¹ * 0.75¹⁴ + 15C2 * 0.25² * 0.75¹³ + 15C3 * 0.25³ * 0.75¹² + 15C4 * 0.25⁴ * 0.75¹¹ + 15C5 * 0.25⁵ * 0.75¹⁰ + 15C6 * 0.25⁶ * 0.75⁹

                = 0.9434

b) P(1 < Y < 5) = P(Y = 2) + P(Y = 3) + P(Y = 4)

                        = 15C2 * 0.25² * 0.75¹³ + 15C3 * 0.25³ * 0.75¹² + 15C4 * 0.25⁴ * 0.75¹¹

                        = 0.6063