Question

A boy tries to message his friend. Each time, the chance the message is delivered is \( \frac{1}{6} \), and the chance it fails is \( \frac{5}{6} \). He sends 6 messages. Find the probability that exactly 5 messages are delivered.

• A. \( \frac{1}{6} \)
• B. \( \frac{5}{6} \)
• C. \( \binom{6}{5} \left( \frac{1}{6} \right)^5 \left( \frac{5}{6} \right) \)
• D. \( \frac{5}{36} \)

Hint:

When solving binomial probability problems, remember to use the binomial distribution formula: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] where \( p \) is the probability of success and \( q \) is the probability of failure.