Question

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.

Hint:

For binomial probability, sum probabilities for required outcomes; use \( \binom{n}{k} p^k q^{n-k} \).

Answer 1

For a fair die, P(odd) = \( \frac{3}{6} = \frac{1}{2} \), P(even) = \( \frac{1}{2} \). Binomial distribution: \( n = 6 \), \( p = \frac{1}{2} \).
P(at least 5 successes) = \( P(X = 5) + P(X = 6) \): \[ P(X = k) = \binom{6}{k} \left( \frac{1}{2} \right)^k \left( \frac{1}{2} \right)^{6-k} = \binom{6}{k} \left( \frac{1}{2} \right)^6. \] \[ P(X = 5) = \binom{6}{5} \cdot \frac{1}{64} = \frac{6}{64} = \frac{3}{32}, \quad P(X = 6) = \binom{6}{6} \cdot \frac{1}{64} = \frac{1}{64}. \] \[ P(X \geq 5) = \frac{3}{32} + \frac{1}{64} = \frac{6}{64} + \frac{1}{64} = \frac{7}{64}. \] Answer: \( \frac{7}{64} \).