16. Inquiry/ Problem Solving A committee is
formed by randomly selecting from eight
nurses and two doctors. What is the
minimum committee size that ensures at
least a 90% probability that it will not be
comprised of nurses only?


We are tasked with finding the minimum size of a committee that ensures a 90% probability that it is **not comprised solely of nurses**, given that the committee is formed by randomly selecting from **eight nurses and two doctors**.

### Step-by-Step Solution:

1. **Total number of people**: There are 8 nurses and 2 doctors, making a total of 10 people.

2. **Nurse-only committee probability**: The goal is to find the minimum committee size \(k\) that ensures a high likelihood that at least one doctor is included. So, we will first compute the probability that the committee consists only of nurses, and then adjust \(k\) until that probability is less than 10%.

The probability that a committee is made up of only nurses can be calculated by considering the total number of ways to select \(k\) people from the 8 nurses, divided by the total number of ways to select \(k\) people from all 10 (nurses + doctors).

   - The number of ways to select \(k\) people from the 8 nurses is: 
     \[
     \binom{8}{k} 
     \]
   - The number of ways to select \(k\) people from the total 10 people (nurses and doctors) is:
     \[
     \binom{10}{k}
     \]
   Therefore, the probability \(P(\text{nurse-only committee})\) is:
     \[
     P(\text{nurse-only committee}) = \frac{\binom{8}{k}}{\binom{10}{k}}
     \]

3. **Condition for 90% certainty**: We want the probability of a nurse-only committee to be less than 10%, meaning:
   \[
   P(\text{nurse-only committee}) < 0.10
   \]
   This ensures a 90% probability that the committee will contain at least one doctor.

4. **Compute for different values of \(k\)**: We now compute the probabilities for different committee sizes \(k\) to determine the minimum value that satisfies the 90% condition.

Let me calculate this.The minimum committee size that ensures at least a 90% probability that the committee will not consist of only nurses is **7**. With a committee of 7 members, the probability that the committee will be comprised of only nurses is approximately 6.67%, which is below the 10% threshold.

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

1. How would the probability change if the total number of people increased?
2. What is the probability that a committee of 6 members consists only of nurses?
3. How would the result change if the number of doctors was increased?
4. Can this method be applied to other random selection problems in different fields?
5. What is the expected number of doctors in a randomly selected committee of 7 members?

**Tip**: When dealing with combinatorics, the binomial coefficient formula \(\binom{n}{k}\) gives the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order.

A committee is formed by randomly selecting from eight nurses and two doctors. What is the minimum committee size that ensures at least a 90% probability that it will not be comprised of nurses only?

This problem explores the minimum committee size required to ensure at least a 90% probability that a committee formed from 8 nurses and 2 doctors will not consist solely of nurses. Key concepts include combinatorics and probability calculations using binomial coefficients.

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