### There are 20 questions in a multiple-choice test. Each question has five choices and one correct answer out of these five choices. A student did not study. He/she answers the questions at random. What are the probabilities that he/she makes (i) more than 10-answer right? (ii) exactly eight right? (iii) at most 7 right?

N = 20, p = 0.2, q = 0.8

(0.8)^x (1 – 0.8)^20 – x

(i) More than 10 correct

P(X = 1) = 0.0576, P(X = 2) = 0.1369,…P(X = 11) = 0.0005, P(X = 12) = 0.0001….

0.0005 + 0.0001 = 0.0006

The probability of getting more than 10 correct is 0.0006 or 0.06%.

(ii) Exactly 8 Correct

Px1 = 0.0576, Px2 = 0.1369,…Px8 = 0.0222

Px8 = 0.0222

The probability of getting exactly 8 correct is 0.0222 or 2.22%

(iii) at most 7 right?

1 – P(X >= 8) =

1 – 0.0322 = 0.9678

The probability of getting at most 7 correct is 0.9678 or 96.78%