Exam 2 Review with key Math 166
1. A group consists of 3 girls and 5 boys. Find how many ways each can be done:
a) Choose a subgroup of 2 girls and 3 boys.
b) Line up the 8 people so that no two girls are next to each other.
c) Line up the 8 people so that the girls are together and the boys are together.
2. A box contains 5 identical red balls, 7 identical blue balls and 9 identical green balls.
How many distinguishable ways can they be arranged in a line?
3. A legal firm has 6 lawyers and 3 new cases. How many ways can
a) a different lawyer be assigned to each case?
b) a lawyer be assigned to each case if they do not need to be different?
c) two lawyers be assigned to each case if no lawyer works on more than one case?
4. P(E)=0.5, P(F)=0.6 P(E
F)=0.9 Find
a) P(E
F ) b) P(E
F ) c) P(E
)
5. A fair coin is tossed 7 times and the sequence of heads and tails is recorded. Find
a) P(HHTHTTH) b) P(at least one H) c) P(at most one H) d) P(at least 2 H's)
e) P(at least 6 H's) f) P(at most 5 H's)
6. In a 3- child family, a girl is as likely as a boy at each birth. The children are born at separate times.
a)What is the probability that the first two are boys and the youngest is a girl?
b) What is the probability of getting two boys and one girl?
c) What is the probability of getting at least two boys?
7. An urn contains 5 red, 4 blue and 3 yellow balls. Two are chosen at random. What is the probability of drawing
a) at least one red?
b) at least one blue?
c) at least one red or at least one blue?
d) two balls of the same color?
e) two balls of different colors?
8. Each dorm room at a certain campus holds 4 people. Four friends are each assigned to one of the 20 dorm rooms.
a) What is the probability that at least 2 will get the same dorm room?
b) What is the probability they will all get the same room?
9. A student taking a 12 question true-false test is told that exactly 4 are true.
If he guesses and answers exactly 4 questions true what is the probability:
a) he will guess the right 4?
b) he will guess exactly three of the right 4?
c) he will guess 3 or 4 of the right 4?
10. a) Seven boys and 4 girls stand in a line. What is the probability that no two girls are next to each other?
b) Seven blue balls and 4 white balls are placed in a line. What is the probability that no two white balls are next to each other?
11. A person tosses a fair coin 4 times. a) Find the probability that the first toss was heads given there were exactly two heads. b) Are the events “the first toss is heads” and “there are exactly two heads” independent?
c) Repeat a) and b) with the coin tossed 5 times.
12. P(A)=0.7 , P(B)=0.6 and P(A
B)=0.2 Are A and B independent? Show why or why
not.
13. In a town in the 1920’s , 70% of the voters were farmers, 20% were merchants and 10% had some other occupation. Assume no one had two occupations. In the mayoral election, 60% of the farmers voted for the democrat, 30% of the merchants voted for the democrat and 70% of the others voted for the democrat. The rest voted republican. Find the probability that
a) a farmer voted democrat.
b) someone who voted democrat was a farmer.
c) someone voted democrat and was a farmer.
14. A lab test tests for use of a certain drug. It is estimated that 8% of people taking the test uses the drug. The test is positive in 4% of those who do not use the drug and is positive in 93% of those who do.
a) Find the probability that someone who gets a positive test actually uses the drug.
b) Find the probability someone who gets two positive consecutive tests uses the drug.
15. Some mice were tested to see if listening to music improved their intelligence. Seventy five mice listened to classical music daily and 75 listened to no music. Eighty of the mice were able to finish a certain maze in less than 5 minutes. Sixty of these had listened to classical music. Are listening to classical music and finishing the maze in less than 5 minutes independent? Use probabilities to support your answer. As a note of interest, some other mice listened to heavy metal music daily. They were bumping into walls and couldn’t finish the maze in 10 minutes.
16. In a voter survey 1500 respondents were asked if they had an immediate family member in the military and if they were voting for the democrat or the republican. The results were that 500 had a family member in the military and 140 of these were voting for the democrat. Overall 660 voted for the democrat and 840 voted republican. Let M be the set of all those who had a family member in the military, R those who voted republican, and D those who voted democrat. Find the probability that
a) someone who had a military relative voted republican.
b) someone who voted democrat does not have a military relative.
c) a voter has no military relative and is voting republican.
d) Are M and R independent?
e) Are M and D independent?
17. Three workers work independently to solve a problem. The events that each solves it are independent. The probabilities that worker 1, 2 or 3 will solve it are 0.7, 0.5, and 0.8 respectively. Find the probability that
a) at least one will solve it.
b) at least one will not solve it.
c) exactly one will solve it.
Key:
1. a) 30 b) 14400 c) 1440
2. 232792560
3. a) 120 b) 216 c) 90
4. a) 0.2 b) 0.4 c) 0.1
5. a) 1/128 b) 127/128 c) 8/128 d) 120/128 e) 8/128 f) 120/128
6. a) 1/8 b) 3/8 c) ˝
7. a) 45/66 b) 38/66 c) 63/66 d) 19/66 e) 47/66
8. a) 0.27325 b) 0.000125
9. a) 1/495 b) 4/495 c) 5/495
10. a) 7/33 b) 7/33
11. a) 0.5 b) yes c) 0.4, not independent
12. a) P(
)=0.5 but P(A)P(B)=0.42
13. a) 0.6 b) 42/55 c) 0.42
14. a) 0.6691 b) 0.9792
15. not independent
16. a) 0.72 b) 0.7879 c) 0.32 d) not independent e) not independent
17. a) 0.97 b) 0.72 c) 0.22