Suppose a drug has a 26% chance of making a person drowsy. Out of a sample of 82 people who each take the drug, what is the probability that no more than 10 of them experience drowsiness, to an exact figure

To solve this problem, we can use the binomial probability formula. The binomial probability formula is:

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

Where:
- \( n \) is the number of trials (in this case, 82 people),
- \( k \) is the number of successes (in this case, the number of people who experience drowsiness),
- \( p \) is the probability of a single success (26% or 0.26),
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

However, we want to find the probability that **no more than 10 people** experience drowsiness, which means we need to find the cumulative probability for \( X \) ranging from 0 to 10.

So, we need to calculate:

\[
P(X \leq 10) = \sum_{k=0}^{10} \binom{82}{k} \cdot 0.26^k \cdot (1-0.26)^{82-k}
\]

This can be calculated using computational methods, as doing it by hand would be very complex.

I'll calculate this for you.The probability that no more than 10 people out of 82 experience drowsiness after taking the drug is approximately **0.001833**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the probability that exactly 10 people experience drowsiness?
2. How would the probability change if the sample size was increased to 100 people?
3. What is the expected number of people who would experience drowsiness in this scenario?
4. How would the result change if the probability of drowsiness increased to 40%?
5. What is the probability that more than 20 people experience drowsiness?

**Tip:** When dealing with binomial distributions, the cumulative distribution function (CDF) is a useful tool for finding the probability of a range of outcomes.

Calculate the probability that no more than 10 people out of 82 experience drowsiness after taking a drug with a 26% chance of causing drowsiness. Explore binomial distribution and probability calculations in detail.

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