Question

In a binomial distribution \( B(n = 10, p) \), the probability of getting exactly 4 successes equals the probability of getting exactly 6 successes. What is the mean of the distribution?

• A. 0.5
• B. 3.5
• C. 2.5
• D. 5

Hint:

In a binomial distribution, the mean is given by \( \mu = n \cdot p \), where \( n \) is the number of trials and \( p \) is the probability of success on a single trial.

Answer 1

The Correct Option is D

In a binomial distribution \( B(n, p) \), the probability of getting exactly \( k \) successes is given by the formula: \[ P(k) = \binom{n}{k} p^k (1 - p)^{n - k} \] Given that the probability of getting exactly 4 successes equals the probability of getting exactly 6 successes, we can set up the following equation: \[ \binom{10}{4} p^4 (1 - p)^6 = \binom{10}{6} p^6 (1 - p)^4 \] By simplifying and solving for \( p \), we find: \[ p = 0.5 \] The mean \( \mu \) of a binomial distribution is given by: \[ \mu = n \cdot p \] Substituting \( n = 10 \) and \( p = 0.5 \), we get: \[ \mu = 10 \cdot 0.5 = 5 \] Thus, the mean of the distribution is \( 5 \).