Key
Exam 2
Math 105 B
Statistics
Dr. Priddy
1. Ann, Mary, Bob and Dave all volunteer to be subjects for a psychology experiment. One person is to be
chosen at random from this group. Each person is equally likely to be chosen.
(a) Describe the sample space for this problem.
Sample Space : Ann, Mary, Bob, Dave
(b) What is the probability that Bob will be chosen?
1 out of 4: 1/4
(c) What is the probability that Bob or Dave will be chosen?
P(A or B) = P(A) +P(B) – P(A and B) = ¼ + ¼ - 0 = 1/2
(d) What is the probability that a woman will be chosen?
There are two women. Therefore, P(2 women) = 2/4 = 1/2
(e) What is the probability that Ann will not be chosen?
There are three ways for Ann not to be chosen. P(Ann not chosen) = ¾ = 1 – P(Ann chosen) = 1 – ¼ = 3/4
2. A coin is to be tossed 1,000 times. What is the probability that the 785th toss is heads?
A. 0
B. 1/4
C. 1/2
D. 1/3
E. 1
3. Andrea did a survey in her statistics class and found that 18 students out of 30 use the library at least twice a
week.
(a) What is the probability that a student chosen at random from the class uses the library at least twice a week?
18/30
(b) What is the probability that a student chosen at random from the class uses the library less often than twice a
week?
1 – 18/30 = 12/30
(c) If two students are chosen at random from the class what is the probability that both of them use the library
at least twice a week?
(18/30) (17/29) = 306/870 = 51/145 (about .3517)
(d) What is the probability that neither of them uses the library at least twice a week?
(12/30)(11/29) = 132/870 = 22/145 (about .1517)
4. A football team’s coach predicts that his team will win 9 out of 10, or 90%, of its games. Which method did
he most likely use to make this prediction?
A. relative frequency
B. intuition
C. law of large numbers
D. equally likely outcomes
E. probability
5. Out of 1,000 car-door handles tested, 23 were found to be defective.
(a) Use the probability formula for relative frequency to determine the probability that a car-door handle is
defective.
f/n = 23/1000
(b) Using the result from part (a), how many car-door handles out of 5,000 do you expect to be defective?
23/1000 * 5000 = 115
6. You draw one card from a standard deck of 52 cards.
(a) What is the probability that the card is the ace of spades?
1/52
(b) What is the probability that the card is a spade? 13/52
(c) What is the probability that the card is an ace or a spade?
P(Ace or Spade) = P(Ace) + P(Spade) – P(Ace And Spade) = 4/52 + 13/52 – 1/52 = 16/52 (about .3077)
7. You draw two cards from a standard deck of 52 cards and do not replace the first card before you draw the
second.
(a) What is the probability that the first card is the king of diamonds and the second is the king of hearts?
(1/52)(1/51) = 1/2652
(b) What is the probability that the first card is the king of diamonds and the second card is a diamond?
(1/52)(12/51) = 12/2652
(c) What is the probability that the first card is not a king and the second card is also not a king?
(48/52)(47/51) = 2256/2652
8. You roll two fair dice, one red and one green.
(a) What is the probability of getting a number less than 5 on both?
16/36
(b) What is the probability of getting a sum of 9 on the two dice?
4/36
(c) What is the probability of getting a five on both?
1/36
9. Ten equally qualified people apply for a job. Three will be selected to interview. How many different
ways of interviewing three people are there?
10 C 3 = 120
10. In a market survey a random sample of 100 people were asked two questions. Did they buy Sparkle
toothpaste last month, and did they see an ad for Sparkle on TV last month? The responses are in the
accompanying table.
Bought Sparkle Did not buy Sparkle Row total
Saw ad
Did not see ad
Column total
32
17
49
33
18
51
65
35
100
For a person selected at random from the sample:
(a) Find the probability that a person saw the ad. 65 saw the ad out of 100 ; 65/100
(b) Find the probability that a person bought Sparkle. 49/100
(c) Find the probability that the person bought Sparkle, given that she or he saw the ad.
65 saw the ad, out of these, 32 bought Sparkle ; 32/65
(d) Find the probability that a person saw the ad and bought Sparkle. 32/100
11. In purchasing a sound system you have 4 choices for speakers, another 2 choices for receivers and 5 choices
for CD players. How many different systems can you construct consisting of 2 identical speakers, one
receiver and 1 CD player? 4 * 2 * 5 = 40
12. Five black and five white marbles are in a box. Without looking, three marbles are taken from the box
without replacement. What is the probability that all three are black? (5/10)(4/9)(3/8)
A. 25/144
B. 1/12
C. 1/8
D. 3/50
E. 1/10
13. Ten horses are entered in a horse race. The 1st, 2nd and 3rd place winners are posted. How many different
postings are possible? Order matters here. 10 P3 = 720
14. (a) Eight students wish to select a subcommittee of four to talk to the president of the college about tuition
costs. In how many ways can the subcommittee be formed.
Order does not matter. “Groups” are being counted.
8
C 4 = 70
(b) There are twelve members of the campus botany club. How many ways can you form a slate of
candidates for the four offices of the organization?
Order matters here due to the offices.
12
P4 = 11,880
15. The dean of women at Brookfield College found that 20% of the female students are majoring in
engineering. If 53% of the students at Brookfield are women what is the probability that a Brookfield
student chosen at random will be a woman engineering major?
P(women in engineering) = (.20)(.53) = .106
16. Which statement below is not a basic probability rule for events A and B?
A. P(entire sample space) = 1
B. For any event A: 0  P( A)  1
C. Events A and B are mutually exclusive if P(A and B) = 1
D. Complement of A: P(not A) = 1 – P(A)
E. Events A and B are independent events if P(A) = P(A, given B)
17. The manager of a water-sports store has found that 5% of her customers purchase sailboats and 15% of her
customers purchase boat trailers. However the probability that a customer will purchase a boat trailer given
that she or he has purchased a sailboat is 80%.
(a) Find the probability that a customer will purchase a sailboat and trailer.
The probability that a customer will purchase a sailboat is 5%. But, if they buy a sailboat, the probability of
them buying a trailer is 80%!
(.05)(.80) = .04
(b) Find the probability that a customer will purchase a sailboat or a trailer.
P(A and B) = P(A) + P(B) – P(A and B) = .05 + .15 - .04 = .16
18. How many ways can you have 5 work days and 2 days off in one week.
There are seven days in the week, choose 5 to work 7 C 5 = 21
Or
There are seven days in a week, choose 2 to be off. 7 C 2 = 21