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.

Answer 1

The Correct Option is C

This problem follows a binomial distribution because we are dealing with a series of independent trials (each message being delivered or not) with two possible outcomes: success (message delivered) or failure (message not delivered). In this case: - The probability of success (a message is delivered) is \( p = \frac{1}{6} \), - The probability of failure (a message is not delivered) is \( q = 1 - p = \frac{5}{6} \), - The number of trials (messages sent) is \( n = 6 \), - We are asked to find the probability of exactly 5 successes (i.e., 5 messages delivered).
Step 1: Apply the binomial distribution formula The formula for the probability of exactly \( k \) successes in \( n \) trials in a binomial distribution is: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] where: - \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes out of \( n \) trials, - \( p^k \) is the probability of \( k \) successes, - \( q^{n-k} \) is the probability of \( n-k \) failures.
Step 2: Substitute values into the formula In this case, we are looking for exactly 5 successes (5 messages delivered), so \( k = 5 \), \( n = 6 \), \( p = \frac{1}{6} \), and \( q = \frac{5}{6} \). Substituting these values into the binomial formula: \[ P(X = 5) = \binom{6}{5} \left( \frac{1}{6} \right)^5 \left( \frac{5}{6} \right)^{6-5} \] \[ P(X = 5) = \binom{6}{5} \left( \frac{1}{6} \right)^5 \left( \frac{5}{6} \right) \]
Step 3: Simplify the expression The binomial coefficient \( \binom{6}{5} \) is equal to 6 (since choosing 5 successes out of 6 trials is the same as choosing 1 failure out of 6 trials): \[ P(X = 5) = 6 \left( \frac{1}{6} \right)^5 \left( \frac{5}{6} \right) \] \[ P(X = 5) = 6 \times \frac{1}{7776} \times \frac{5}{6} \] \[ P(X = 5) = \frac{6 \times 5}{7776 \times 6} = \frac{30}{7776} = \frac{5}{1296} \] Thus, the probability that exactly 5 messages are delivered is \( \boxed{\frac{5}{1296}} \).