Theory question 112
Probability(a) In a school, the ratio of those who passed to those who failed in a History test is 4 : 1. If 7 students are selected at random from the school, find, correct to two decimal places, the probability that: (i) at least 3 students passed the test: (ii) between 3 and 6 students failed the test. (b) A fair die is thrown five times, find the probability of obtaining a six three times.
Show the model answer
Model answer
P(pass) = 0.8, P(fail) = 0.2, n = 7.
(a)(i) 1 − P(0, 1 or 2 pass) = 1 − 0.0047 ≈ 1.00 (0.9953).
(ii) Between 3 and 6 (4 or 5 failed): ⁷C₄(0.2)⁴(0.8)³ + ⁷C₅(0.2)⁵(0.8)² ≈ 0.0287 + 0.0043 = 0.03. (Including 3 and 6: ≈ 0.15.)
(b) ⁵C₃(1/6)³(5/6)² = 250/7776 = 125/3888 ≈ 0.032.
Explanation
Use the binomial distribution with the right probability for each event.