To solve the given problems involving a binomial distribution, we need to use the binomial probability formula and cumulative distribution function (CDF).

Given:

• The probability of success ($p$) = 0.57
• The number of trials ($n$) = 15

The binomial probability formula for exactly $k$ successes is given by: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

where $\binom{n}{k}$ is the binomial coefficient: $\binom{n}{k} = \frac{n!}{k!(n - k)!}$

a) Getting exactly 13 successes

Using the formula: $P(X = 13) = \binom{15}{13} (0.57)^{13} (0.43)^{2}$

First, calculate