Suppose a drug has a 24% chance of making a person drowsy. Out of a sample of 79 people who each take the drug, what is the probability that no more than 10 of them experience drowsiness?

This problem involves finding the probability that no more than 10 people out of 79 experience drowsiness when the probability of any single person experiencing drowsiness is 24%. The situation can be modeled using the binomial distribution.

### Step 1: Define the Parameters
- \( n = 79 \) (number of trials or people)
- \( p = 0.24 \) (probability of success, i.e., a person experiencing drowsiness)
- \( X \) represents the number of people who experience drowsiness, so \( X \) follows a binomial distribution \( X \sim \text{Binomial}(n = 79, p = 0.24) \).

We need to calculate \( P(X \leq 10) \), which is the probability that no more than 10 people experience drowsiness.

### Step 2: Use the Binomial Formula
The probability mass function for a binomial distribution is given by:

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

where \( \binom{n}{k} \) is the binomial coefficient.

However, summing this up for \( k = 0 \) to \( k = 10 \) can be tedious manually, so we typically use statistical software, a calculator, or an online tool to compute the cumulative probability.

### Step 3: Compute \( P(X \leq 10) \)
I will compute this using Python.

The probability that no more than 10 people out of 79 experience drowsiness is approximately \( 0.0094 \), or 0.94%.

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the probability that exactly 10 people experience drowsiness?
2. What is the expected number of people who will experience drowsiness?
3. What is the standard deviation for the number of people who experience drowsiness?
4. How would the probability change if the sample size were 100 instead of 79?
5. What is the probability that more than 20 people experience drowsiness?

**Tip:** When dealing with binomial distributions, cumulative probabilities are often easier to compute using software tools or tables instead of manual calculations.

Explore the probability calculation where no more than 10 out of 79 people experience drowsiness after taking a drug with a 24% chance of causing drowsiness. Learn how to apply binomial distribution in this scenario.

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