To calculate the probability of exactly 4 successes in a binomial experiment, we can use the binomial probability formula:

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n-k}$

Where:

• $n = 5$ (number of trials)
• $k = 4$ (number of successes)
• $p = 0.65$ (probability of success)
• $1 - p = 0.35$ (probability of failure)

Step 1: Calculate the binomial coefficient $\binom{5}{4}$

$\binom{n}{k} = \frac{n!}{k!(n - k)!} = \frac{5!}{4!(5 - 4)!} = \frac{5!}{4!1!} = 5$

Step 2: Apply the binomial formula

$P(X = 4) = 5 \cdot (0.65)^4 \cdot (0.35)^1$

Let me calculate that.The probability of getting exactly 4 successes in the binomial experiment is $0.312$, rounded to three decimal places.

Would you like further clarification or details on any part of this solution? Here are some related questions you might explore:

1. What is the probability of getting exactly 5 successes in this experiment?
2. How would the probability change if the number of trials increased to 6?
3. What is the expected number of successes in this binomial experiment?
4. How do we calculate the probability of getting fewer than 4 successes?
5. How would you calculate the cumulative probability of 4 or fewer successes?

Tip: In a binomial distribution, the mean is given by $\mu = np$, and the variance by $\sigma^2 = np(1-p)$.