On the freshman-sophomore board, there are 16 freshman members and 13 sophomore board members. If a committee of 12 members is randomly chosen, what is the probability that the committee has exactly 6 freshman?
Round your answer to 6 decimal places as needed.

The problem can be solved using combinations, as it involves selecting a certain number of freshman and sophomore members from the board. We are tasked with finding the probability that a committee of 12 members has exactly 6 freshman members.

### Step-by-step solution:

1. **Total number of ways to choose 12 members**:
   - There are 16 freshman and 13 sophomores, for a total of 16 + 13 = 29 board members.
   - The total number of ways to select 12 members from 29 is given by the combination formula:
   
   \[
   \text{Total ways} = \binom{29}{12} = \frac{29!}{12!(29-12)!}
   \]

2. **Number of favorable ways (exactly 6 freshman)**:
   - To have exactly 6 freshmen, we need to choose 6 from the 16 freshman and 6 from the 13 sophomores.
   - The number of ways to choose 6 freshman from 16 is:
   
   \[
   \text{Ways to choose 6 freshman} = \binom{16}{6} = \frac{16!}{6!(16-6)!}
   \]
   
   - The number of ways to choose 6 sophomores from 13 is:
   
   \[
   \text{Ways to choose 6 sophomores} = \binom{13}{6} = \frac{13!}{6!(13-6)!}
   \]

3. **Probability calculation**:
   - The probability of selecting exactly 6 freshman and 6 sophomores is the ratio of favorable outcomes to total outcomes:
   
   \[
   \text{Probability} = \frac{\binom{16}{6} \times \binom{13}{6}}{\binom{29}{12}}
   \]

Let me calculate this now.The probability that a randomly chosen committee of 12 members includes exactly 6 freshman members is approximately \(0.264794\) (rounded to six decimal places).

Would you like more details or have any other questions?

Here are 5 related questions you might find interesting:
1. What is the probability of selecting exactly 7 freshman members?
2. What is the probability of selecting only sophomores on the committee?
3. How many ways are there to form a committee of 10 members with an equal number of freshmen and sophomores?
4. What is the probability of selecting at least one freshman member?
5. What is the probability of selecting fewer than 5 freshman members?

**Tip:** When calculating probabilities involving combinations, understanding how to use the combination formula efficiently is crucial for solving complex problems quickly!

On the freshman-sophomore board, there are 16 freshman members and 13 sophomore board members. If a committee of 12 members is randomly chosen, what is the probability that the committee has exactly 6 freshman? Round your answer to 6 decimal places as needed.

Learn how to calculate the probability of selecting a committee of 12 members with exactly 6 freshmen from a board of 16 freshman members and 13 sophomore members. This problem uses combinatorics and probability formulas and is suitable for students in Grades 11-12 or introductory college courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation