Theory question 113
Probability(a) A bag contains 16 identical balls of which 4 are green. A boy picks a ball at random from the bag and replaces it. If this is repeated 5 times, what is the probability that he: (i) did not pick a green ball; (ii) picked a green ball at least three times?
(b) The deviations from a mean of values from a set of data are −2, (m − 1), (m2 + 1), −1, 2, (2m − 1) and −2. Find the possible values of m.
Show the model answer
Model answer
(a) p(green) = 1/4, q = 3/4, n = 5.
(i) (3/4)⁵ = 243/1024 ≈ 0.237.
(ii) P(3) + P(4) + P(5) = 10(1/4)³(3/4)² + 5(1/4)⁴(3/4) + (1/4)⁵ = (90 + 15 + 1)/1024 = 53/512 ≈ 0.104.
(b) Deviations from the mean add up to 0: m² + 3m − 4 = 0, (m + 4)(m − 1) = 0, so m = 1 or m = −4.
Explanation
Deviations from the mean always sum to zero.