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The problem you're asking about involves a binomial distribution \( X \sim B(n=15, p=\frac{3}{4}) \). We need to solve two parts:

(a) \( P(X = 8) \)  
(b) \( P(2 \leq X < 6) \)

Let's solve these step by step.

### (a) \( P(X = 8) \)

For a binomial distribution, the probability of getting exactly \( X = k \) successes in \( n \) trials is given by the binomial probability formula:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n = 15 \) is the number of trials,
- \( p = \frac{3}{4} \) is the probability of success on a single trial,
- \( k = 8 \) is the number of successes.

We can compute:

\[
P(X = 8) = \binom{15}{8} \left( \frac{3}{4} \right)^8 \left( \frac{1}{4} \right)^{15-8}
\]

Now let's compute that.

### (b) \( P(2 \leq X < 6) \)

This is asking for the cumulative probability for \( X \) between 2 and 5 inclusive. So we need to calculate:

\[
P(2 \leq X < 6) = P(2 \leq X \leq 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]

We will calculate each of these probabilities using the binomial formula from part (a) and sum them up.

Let me now compute both parts (a) and (b).

If X ~ B(15, 3/4), find: (a) P(X=8), (b) P(2 ≤ X < 6).

Solve a binomial distribution problem where X follows B(15, 3/4). Find P(X = 8) using the binomial probability formula, and calculate cumulative probabilities for P(2 ≤ X < 6). Suitable for students in Grades 10-12.

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