The problem asks for the probability that at least 10 out of 15 people recover from a disease, given that the recovery probability is 0.4. This situation can be modeled using a binomial distribution.

The binomial distribution has the following parameters:

• $n = 15$ (the number of trials, i.e., people),
• $p = 0.4$ (the probability of success, i.e., recovery),
• $X$ (the random variable representing the number of successes, i.e., people who recover).

We are looking for $P(X \geq 10)$, the probability that at least 10 people recover.

This can be calculated as:

$P(X \geq 10) = 1 - P(X < 10)$

The formula for the binomial probability mass function is:

$P(X = x) = \binom{15}{x} p^x (1-p)^{15-x}$

Now, let's analyze the answer options based on this.

• Option (a) uses the correct complement formula, but with incorrect probabilities.

  • It uses $0.6$ as the probability of recovery instead of $0.4$.

• Option (b) directly calculates $P(X \geq 10)$ with the correct probability values $0.4$ and $0.6$, but doesn't use the complement method. This approach is valid, but more cumbersome to compute.

• Option (c) is similar to (a), but it uses the correct probabilities $0.4$ for recovery and $0.6$ for non-recovery, making it the correct option.

• Option (d) uses incorrect probabilities again, similar to (a), but does not employ the complement.

Correct Answer:

• (c) is the correct option because it correctly uses the complement rule and the correct probabilities $p = 0.4$ and $1 - p = 0.6$.

Would you like further clarification or a more detailed explanation?

Follow-up Questions:

1. How do you calculate binomial probabilities for small values of $X$?
2. Can you explain the complement rule in probability?
3. How does changing the probability of success $p$ affect the binomial distribution?
4. What would the probability be if $p$ were 0.6 instead of 0.4?
5. What is the expected value of the binomial distribution in this problem?

Tip:

When dealing with "at least" problems in binomial distributions, it's often easier to calculate the complement, $1 - P(X < k)$.