Question

X denotes the number of heads in $ n $ tosses of a fair coin. If $ P(X = 4),\ P(X = 5),\ P(X = 6) $ are in arithmetic progression, find the largest possible value of $ n $.

• A. 7
• B. 14
• C. 21
• D. 28

Hint:

Use binomial probabilities and cancel common terms. Check for values of \( n \) that satisfy the arithmetic condition.

Answer 1

The Correct Option is B

Let \( X \sim B(n, \frac{1}{2}) \). Then, using binomial formula: \[ P(X = r) = {}^nC_r \left(\frac{1}{2}\right)^n \] Since all terms have \( \left(\frac{1}{2}\right)^n \), the condition becomes: \[ {}^nC_5 - {}^nC_4 = {}^nC_6 - {}^nC_5 \Rightarrow 2{}^nC_5 = {}^nC_4 + {}^nC_6 \] Try \( n = 14 \): \[ {}^{14}C_4 = 1001,\quad {}^{14}C_5 = 2002,\quad {}^{14}C_6 = 3003 \Rightarrow 1001 + 3003 = 4004 = 2 \cdot 2002 \Rightarrow \text{Condition satisfied} \]