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.