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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 \( 1 - p = 0.6 \).

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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, \( 1 - P(X < k) \).

The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that at least 10 survive?

This problem explores binomial distribution and probability, determining the likelihood that at least 10 out of 15 patients recover from a rare disease, with a recovery probability of 0.4. Learn how to apply the complement rule and calculate binomial probabilities using key formulas.

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