A fair coin is tossed three times. Let $X$ denote the number of heads obtained. (i) Write the probability distribution of $X$. (ii) Find the expected value $E[X]$.

Step-by-step solution

1. List all $2^3=8$ equally likely outcomes of three tosses.
2. Count outcomes with 0 heads: only TTT → 1 outcome, so $P(X=0)=1/8$.
3. Count outcomes with 1 head: HTT, THT, TTH → 3 outcomes, so $P(X=1)=3/8$.
4. Count outcomes with 2 heads: HHT, HTH, THH → 3 outcomes, so $P(X=2)=3/8$.
5. Count outcomes with 3 heads: HHH → 1 outcome, so $P(X=3)=1/8$.
6. Compute expectation using $E[X]=\sum xP(X=x)$: $0\cdot \frac{1}{8}+1\cdot \frac{3}{8}+2\cdot \frac{3}{8}+3\cdot \frac{1}{8}=\frac{3}{2}$.

Concept tested: Discrete random variable and expectation

Published 5 September 2026 · How this solution was prepared and checked