Finite Homework, need help: permutations, combinations, probability.

Question

1. For this problem, assume the cooler contains 32 cans, 22 of which are regular, and the rest of which are diet.

What is the probability of selecting 5 cans from the cooler so that at least 4 are diet?

What is the probability of selecting 5 cans from the cooler so that some are diet and some are regular?

 

2. Assume that there are 7 different issues of Time, 6 different issues of Sports Illustrated, and 2 different issues of Newsweek, including the December 1st issue, on the rack. You choose 4 of them at random.

What is the probability that you choose 2 issues of Time and 2 issues of Sports Illustrated?

What is the probability that you choose at least 3 of the Sports Illustrated magazines?

 

3. Assume that the set S has 14 elements.

How many subsets of S have at most 2 elements?

 

4. Assume that the playbook contains 14 passing plays and 9 running plays. The coach randomly selects 8 plays from the playbook.

What is the probability that the coach selects at least 2 passing plays and at least 3 running plays?

 

5. Assume that there are 6 unfilled roles: 4 male and 2 female. There are 8 men and 4 women, including Jennifer, auditioning for a part in the play.

How many different casts are there?

How many of these casts include Jennifer?

 

6. For this problem, assume the bag contains 16 basketballs, 7 of which have defective valves. You choose 3 out of the bag randomly.

What is the probability that at least 1 will have a defective valve?

Answer 1

The following will help you get started on 1.) :

 

1.) Probability of selecting a can of diet from cooler = 10/32

 

Probability of selecting 4 or 5 cans of diet from any 5 random selections = ₁₀P₅ / ₃₂P_{5  *  10}P₄ / ₃₂P_{4 =}

 

[ (10 * 9 * 8 * 7 * 6) / (32 * 31 * 30 * 29 * 28) ]  *  [ (10 * 9 * 8 * 7 * 6 * 5) / (32 * 31 * 30 * 29) ]  =

 

30240/24165120 * 151200/863040  = 

 

0.00125 * 0.17519  =  0.00022

 

Total Probability of this occurrence = 10/32 * 0.00022 = 0.00006875