CHAPTER 2 - Probability
SHORT ANSWER
1. Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider
observing the direction for each of three successive vehicles.
a. List all outcomes in the event A that all three vehicles go in the same direction. b.
List all outcomes in the event B that all three vehicles take different directions. c. List
all outcomes in the event C that exactly two of the three vehicles turn right.
d. List all outcomes in the even D that exactly tow vehicles go in the same direction.
e. List outcomes in ,C D, and C D.
ANS:
a. Event A = {RRR, LLL, SSS}
b. Event B = {RLS, RSL, LRS, LSR, SRL, SLR}
c. Event C = {RRL, RRS, RLR, RSE, LRR, SRR}
d. Event D = {RRL, RRS, RLR, RSE, LRR, SRR, LLR, LLS, LRL, LSL, RLL, SLL, SSR, SSL, SRS, SLS, RSS,
LSS}
e. Event contains outcomes where all cars go the same direction, or they all go different directions:
= {RRR, LLL, SSS, RLS, RSL, LRS, LSR, SRL, SLR}
Because Event D totally encloses Event C, the compound event C D = D. Therefore, C D = {RRL, RRS,
RLR, RSR, LRR, SRR, LLR, LLS, LRL, LSL, RLL, SLL, SSR, SSL, SRS, SLS, RSS, LSS}
Using similar reasoning, we see that the compound event C D = C. Therefore, C D = {RRL, RRS, RLR,
RSR, LRR, SRR}
PTS: 1
2. Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V).
a. What are the 16 outcomes in S?
b. Which outcomes are in the event that exactly two of the selected mortgages are fixed rate?
c. Which outcomes are in the event that all four mortgages are of the same type?
d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage?
e. What is the union of the events in parts (c) and (d), and what is the intersection of these two
events?
f. What are the union and intersection of the two events in parts (b) and (c)?
, ANS:
a.
Home Mortgage Number
Outcome 1 2 3 4
1 F F F F
2 F F F V
3 F F V F
4 F F V V
5 F V F F
6 F V F V
7 F V V F
8 F V V V
9 V F F F
10 V F F V
11 V F V F
12 V F V V
13 V V F F
14 V V F V
15 V V V F
16 V V V V
b. Outcome numbers 4, 6, 7, 10, 11,13
c. Outcome numbers 1, 16
d. Outcome numbers 1, 2, 3, 5, 9
e. The Union: outcomes 1, 2, 3, 5, 9, 16. The Intersection: outcome 1.
f. The Union: outcomes 1, 4, 6, 7, 10, 11, 13, 16. The Intersection: this cannot happen. (There are no outcomes
in common) :b c= .
PTS: 1
3. A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other
three (3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a
second printing has been selected. One possible outcome is 4, and another is 125.
a. List the S.
b. Let A denote the event that exactly one book must be examined. What outcomes are in A?
c. Let B be the event that book 4 is the one selected. What outcomes are in B?
d. Let C be the event that book 2 is not examined. What outcomes are in C?
ANS:
a.
Outcome Number Outcome
1 123
2 124
3 125
4 213
5 214
6 215
7 13
, 8 14
9 15
10 23
11 24
12 25
13 3
14 4
15 5
b. Outcome numbers 13, 14, 15, so A={3, 4, 5}
c. Outcome numbers 2, 5, 8, 11, 14 so B={124, 214, 14, 24, 4}
d. Outcome numbers 7, 8, 9, 13, 14, 15 so C={13, 14, 15, 3, 4, 5}
PTS: 1
4. The Department of Statistics at a state university in California has just completed voting by secret ballot for a
department head. The ballot box contains four slips with votes for candidate A and three slips with votes for
candidate B. Suppose these slips are removed from the box one by one.
a. List all possible outcomes.
b. Suppose a running tally is kept as slips are removed. For what outcomes does A remain ahead of B throughout
the tally?
ANS:
a. S = {BBBAAAA, BBABAAA, BBAABAA, BBAAABA, BBAAAAB, BABBAAA,
BABABAA, BABAABA, BABAAAB, BAABBAA, BAABABA, BAABAAB, BAAABBA, BAAABAB,
BAAAABB, ABBBAAA, ABBABAA, ABBAABA, ABBAAAB, ABABBAA, ABABABA, ABABAAB,
ABAABBA, ABAABAB, ABAAABB, AABBBAA, AABBABA, AABBAAB, AABABBA, AABABAB,
AABAABB, AAABBBA, AAABBAB, AAABABB, AAAABBB}
b. {AAAABBB, AAABABB, AAABBAB, AABAABB, AABABAB}
PTS: 1
5. Let A denote the event that the next item checked out at a college library is a math book, and let B be the event that
the next item checked out is a history book. Suppose that P(A) = .40 and P(B) = .50.
a. Why is it not the case that P(A) + P(B) = 1?
b. Calculate P( )
c. Calculate P(A B).
d. Calculate P( ).
ANS:
a. The probabilities do not add to 1 because there are other items besides math and history books to
be checked out from the library.
b. P( ) = 1 – P(A) = 1 - .40 = .60
c. P(A B) = P(A) + P(B) = .40 + .50 = .90 (since A and B are mutually exclusive events)
d. P( ) = P[ ] (De Morgan’s law) = 1 – P(A B) = 1 - .90 = .10
PTS: 1
6. A large company offers its employees two different health insurance plans and two different dental insurance plans.
Plan 1 of each type is relatively inexpensive, but restricts the choice of providers, whereas plan 2 is more expensive
but more flexible. The accompanying table gives the percentages of employees who have chosen the various plans:
SHORT ANSWER
1. Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider
observing the direction for each of three successive vehicles.
a. List all outcomes in the event A that all three vehicles go in the same direction. b.
List all outcomes in the event B that all three vehicles take different directions. c. List
all outcomes in the event C that exactly two of the three vehicles turn right.
d. List all outcomes in the even D that exactly tow vehicles go in the same direction.
e. List outcomes in ,C D, and C D.
ANS:
a. Event A = {RRR, LLL, SSS}
b. Event B = {RLS, RSL, LRS, LSR, SRL, SLR}
c. Event C = {RRL, RRS, RLR, RSE, LRR, SRR}
d. Event D = {RRL, RRS, RLR, RSE, LRR, SRR, LLR, LLS, LRL, LSL, RLL, SLL, SSR, SSL, SRS, SLS, RSS,
LSS}
e. Event contains outcomes where all cars go the same direction, or they all go different directions:
= {RRR, LLL, SSS, RLS, RSL, LRS, LSR, SRL, SLR}
Because Event D totally encloses Event C, the compound event C D = D. Therefore, C D = {RRL, RRS,
RLR, RSR, LRR, SRR, LLR, LLS, LRL, LSL, RLL, SLL, SSR, SSL, SRS, SLS, RSS, LSS}
Using similar reasoning, we see that the compound event C D = C. Therefore, C D = {RRL, RRS, RLR,
RSR, LRR, SRR}
PTS: 1
2. Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V).
a. What are the 16 outcomes in S?
b. Which outcomes are in the event that exactly two of the selected mortgages are fixed rate?
c. Which outcomes are in the event that all four mortgages are of the same type?
d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage?
e. What is the union of the events in parts (c) and (d), and what is the intersection of these two
events?
f. What are the union and intersection of the two events in parts (b) and (c)?
, ANS:
a.
Home Mortgage Number
Outcome 1 2 3 4
1 F F F F
2 F F F V
3 F F V F
4 F F V V
5 F V F F
6 F V F V
7 F V V F
8 F V V V
9 V F F F
10 V F F V
11 V F V F
12 V F V V
13 V V F F
14 V V F V
15 V V V F
16 V V V V
b. Outcome numbers 4, 6, 7, 10, 11,13
c. Outcome numbers 1, 16
d. Outcome numbers 1, 2, 3, 5, 9
e. The Union: outcomes 1, 2, 3, 5, 9, 16. The Intersection: outcome 1.
f. The Union: outcomes 1, 4, 6, 7, 10, 11, 13, 16. The Intersection: this cannot happen. (There are no outcomes
in common) :b c= .
PTS: 1
3. A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other
three (3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a
second printing has been selected. One possible outcome is 4, and another is 125.
a. List the S.
b. Let A denote the event that exactly one book must be examined. What outcomes are in A?
c. Let B be the event that book 4 is the one selected. What outcomes are in B?
d. Let C be the event that book 2 is not examined. What outcomes are in C?
ANS:
a.
Outcome Number Outcome
1 123
2 124
3 125
4 213
5 214
6 215
7 13
, 8 14
9 15
10 23
11 24
12 25
13 3
14 4
15 5
b. Outcome numbers 13, 14, 15, so A={3, 4, 5}
c. Outcome numbers 2, 5, 8, 11, 14 so B={124, 214, 14, 24, 4}
d. Outcome numbers 7, 8, 9, 13, 14, 15 so C={13, 14, 15, 3, 4, 5}
PTS: 1
4. The Department of Statistics at a state university in California has just completed voting by secret ballot for a
department head. The ballot box contains four slips with votes for candidate A and three slips with votes for
candidate B. Suppose these slips are removed from the box one by one.
a. List all possible outcomes.
b. Suppose a running tally is kept as slips are removed. For what outcomes does A remain ahead of B throughout
the tally?
ANS:
a. S = {BBBAAAA, BBABAAA, BBAABAA, BBAAABA, BBAAAAB, BABBAAA,
BABABAA, BABAABA, BABAAAB, BAABBAA, BAABABA, BAABAAB, BAAABBA, BAAABAB,
BAAAABB, ABBBAAA, ABBABAA, ABBAABA, ABBAAAB, ABABBAA, ABABABA, ABABAAB,
ABAABBA, ABAABAB, ABAAABB, AABBBAA, AABBABA, AABBAAB, AABABBA, AABABAB,
AABAABB, AAABBBA, AAABBAB, AAABABB, AAAABBB}
b. {AAAABBB, AAABABB, AAABBAB, AABAABB, AABABAB}
PTS: 1
5. Let A denote the event that the next item checked out at a college library is a math book, and let B be the event that
the next item checked out is a history book. Suppose that P(A) = .40 and P(B) = .50.
a. Why is it not the case that P(A) + P(B) = 1?
b. Calculate P( )
c. Calculate P(A B).
d. Calculate P( ).
ANS:
a. The probabilities do not add to 1 because there are other items besides math and history books to
be checked out from the library.
b. P( ) = 1 – P(A) = 1 - .40 = .60
c. P(A B) = P(A) + P(B) = .40 + .50 = .90 (since A and B are mutually exclusive events)
d. P( ) = P[ ] (De Morgan’s law) = 1 – P(A B) = 1 - .90 = .10
PTS: 1
6. A large company offers its employees two different health insurance plans and two different dental insurance plans.
Plan 1 of each type is relatively inexpensive, but restricts the choice of providers, whereas plan 2 is more expensive
but more flexible. The accompanying table gives the percentages of employees who have chosen the various plans:
