We are given a binomial distribution $X \sim B(8, 0.7)$, where $n = 8$ and $p = 0.7$. The possible values of the random variable $X$ are $0, 1, 2, 3, 4, 5, 6, 7, 8$. We are asked to find the following probabilities:

Binomial probability mass function:

The formula for a binomial probability is: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where:

• $n$ is the number of trials,
• $k$ is the number of successes,
• $p$ is the probability of success in each trial.

Now, let's compute each probability step by step.

1) $P(X = 4)$

We calculate $P(X = 4)$ using the binomial probability formula: $P(X = 4) = \binom{8}{4} (0.7)^4 (0.3)^4$

2) $P(X \geq 6)$

To find $P(X \geq 6)$, we need to sum the probabilities of $X = 6, 7, 8$: $P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8)$ We will calculate each of these using the binomial formula.

3) $P(X > 5)$

This is the probability that $X$ is greater than 5, which includes $X = 6, 7, 8$: $P(X > 5) = P(X = 6) + P(X = 7) + P(X = 8)$ This is the same as $P(X \geq 6)$, so we will reuse the result from part (2).

4) $P(3 \leq X \leq 7)$

This requires summing the probabilities from $X = 3$ to $X = 7$: $P(3 \leq X \leq 7) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)$

I'll now compute these probabilities.Here are the results for the given probabilities:

1. $P(X = 4) = 0.1361$
2. $P(X \geq 6) = 0.5518$
3. $P(X > 5) = 0.5518$ (same as $P(X \geq 6)$)
4. $P(3 \leq X \leq 7) = 0.9311$

Would you like more details or clarification on any of these steps?

Here are five related questions for further exploration:

1. How do the values of $p$ and $n$ affect the shape of the binomial distribution?
2. What is the expected value and variance of a binomial distribution?
3. How would you calculate $P(X \leq 5)$ for this distribution?
4. What is the cumulative distribution function (CDF) of a binomial distribution, and how is it used?
5. How do you approximate a binomial distribution using a normal distribution for large $n$?

Tip: In binomial problems, cumulative probabilities can often be computed more efficiently using tables or software, especially for larger $n$.