Subtopic: Poisson, normal and binomial distributions
Question Statement:
A fair coin is tossed $6$ times independently. The probability of obtaining exactly $3$ heads is ________ (round off to 3 decimal places).
Answer Key:
0.313
Step-by-Step Solution & Explanation
Let $X$ be the number of heads in $n = 6$ trials with success probability $p = 0.5$.
$X$ follows a Binomial distribution $B(n=6, p=0.5)$:
$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
For $k = 3$:
$P(X = 3) = \binom{6}{3} (0.5)^3 (0.5)^3 = \frac{6!}{3!3!} (0.5)^6 = 20 \times \frac{1}{64} = \frac{20}{64} = \frac{5}{16} = 0.3125$
Rounding off to 3 decimal places gives $0.313$.