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The problem asks to solve three probability questions for a binomial distribution:

The random variable \( X \sim B(12, 0.3) \), which means \( X \) follows a binomial distribution with parameters \( n = 12 \) (the number of trials) and \( p = 0.3 \) (the probability of success in each trial).

The binomial probability mass function (PMF) is given by:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where \( \binom{n}{k} \) is the binomial coefficient.

We are asked to find the following:

### (a) \( P(X \leq 2) \)
This is the cumulative probability for \( X \) being less than or equal to 2:
\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]

### (b) \( P(X < 6) \)
This is the cumulative probability for \( X \) being less than 6:
\[
P(X < 6) = P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]

### (c) \( P(X \geq 7) \)
This is the complement of \( P(X \leq 6) \):
\[
P(X \geq 7) = 1 - P(X \leq 6)
\]

Let me calculate each of these probabilities for you.The probabilities for the binomial distribution \( X \sim B(12, 0.3) \) are:

(a) \( P(X \leq 2) = 0.2528 \)

(b) \( P(X < 6) = 0.8822 \)

(c) \( P(X \geq 7) = 0.0386 \)

Would you like more details on any of the steps or explanations?

Here are five follow-up questions you might consider:
1. What is the probability that \( X = 3 \) for this binomial distribution?
2. How does changing the value of \( p \) affect these probabilities?
3. Can you calculate \( P(X = 5) \) directly?
4. How do you calculate the mean and variance of a binomial distribution?
5. What is the probability of getting more than 4 successes in this binomial experiment?

**Tip**: The binomial cumulative distribution function (CDF) is useful when you want to calculate probabilities for ranges of values, not just individual outcomes.

The random variable X ~ B(12, 0.3). Find: (a) P(X ≤ 2) (b) P(X < 6) (c) P(X ≥ 7)

Solve a probability problem involving a binomial distribution with parameters B(12, 0.3). Learn how to calculate cumulative probabilities for X ≤ 2, X < 6, and X ≥ 7 using the binomial probability theorem. Suitable for students in Grades 10-12.

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