A-Level Discrete random variables: worked solution

Question

X~B(16, p) is the number, out of 16, of pralines above the recommended sugar level. The single most likely value of X is 2. Find the exact interval of values p may take. [3]

X ~ B(16, p), so P(X = k) = ¹⁶C_k p^k (1 − p)^(16−k). "2 is the single most likely value" means P(X = 2) is strictly greater than the probabilities on either side of it, so both of these must hold: P(X = 2) > P(X = 1) and P(X = 2) > P(X = 3). First inequality, using ¹⁶C₂ = 120 and ¹⁶C₁ = 16: 120 p²(1 − p)¹⁴ > 16 p(1 − p)¹⁵ Divide both sides by p(1 − p)¹⁴, which is positive for 0 < p < 1 so the inequality sign does not change: 120p > 16(1 − p) 120p > 16 − 16p 136p > 16 p > 16/136 p > 2/17 Second inequality, using ¹⁶C₃ = 560: 120 p²(1 − p)¹⁴ > 560 p³(1 − p)¹³ Divide both sides by p²(1 − p)¹³, again positive: 120(1 − p) > 560p 120 > 680p p < 120/680 p < 3/17 Both conditions must hold at once, so 2/17 < p < 3/17.