Given a binomial distribution in which the probability of success is 0.57 and the number of trials is 15, what is the probability for each of the following:

(Round your answers to 3 decimal places)

a) Getting exactly 13 successes?

Number

b) Getting more than 13 successes?

Number

c) Getting less than or equal to 13 successes?

Number

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

Explore how to calculate probabilities using the binomial distribution with specific examples. Learn about the binomial probability formula and its application to scenarios involving success rates and trials. Round answers to three decimal places.

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