Probability Answers

Question

A committee of five is selected from a total of 3 freshman, 9 sophomores and 5 juniors.  Find the probabilities of the following:

 

1.  at least three freshman

2.  all five of the same class

3. not all five from the same class

4.  exactly three of the same class

Answer 1

We have to choose 5 persons from among 17. The Total number of committee's that can be formed is thus ¹⁷C₅.

 

1. The number of committee's with at least 3 freshmen is obviously include all 3 freshman (1 way) and 2 out of the 14 sophomores and juniors ¹⁴C₂ thus the probability is (1. ¹⁴C₂)/(¹⁷C₅)

 

2. The number of committee's with all 5 sophomores is ^₉C₅  and those with all 5 Juniors is exactly 1, thus the probability is (⁹C₅ +1)/ ¹⁷C₅

 

3. This would be the complement of the answer in 2. above so it would be 1- (answer in 2)

 

4. This committee can be formed in various ways. First 3 freshman and 2 from the 14 sophomores and juniors, number of ways  is 1. ¹⁴C₂

 

Second 3 sophomores from the 9 and 2 from the 8 freshman and juniors, number of ways  is ⁹C₃. ⁸C₂

 

Third is 3 juniors from the 5 and 2 from the 12 freshman and sophomores, number of ways  is ⁵C₃. ¹²C₂  

 

Now add the results from the 3 cases above and divide by ¹⁷C₅ to get the probability.