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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 \( n \).

Let X ~ B(8, 0.7) and the possible values of the random variable X are 0, 1, 2, 3, 4, 5, 6, 7, 8. Find P(X = 4), P(X >= 6), P(X > 5), P(3 <= X <= 7)

Learn how to solve a binomial distribution problem where X follows a binomial distribution B(8, 0.7). This step-by-step guide covers the calculation of P(X = 4), P(X >= 6), P(X > 5), and P(3 <= X <= 7). Suitable for high school students in Grades 10-12.

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