# Tutorial 1(Solution).pdf

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Ryerson University

Mathematics

MTH 304

Elizabeth Carlson

Winter

Description

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 inD , C D, and C D.
ANSWER:
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 D contains outcomes where all cars go the same direction, or they all go different directions:
D = { 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 }
2. 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 printingsA 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?
ANSWER:
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}
3. A certain sports car comes equipped with either an automatic or a manual transmission, and the car is
available in one of four colors. Relevant probabilities for various combinations of transmission type and
color are given in the accompanying table.
Color
Transmission Type White Blue Black Red
A .13 .10 .11 .11
M .15 .07 .15 .18
Let A = (automatic transmission), B = {black}, and C = {white}.
a. Calculate P(A), P(B), and P(A B).
b. Calculate both P(A|B) and P(B|A), and explain in context what each of these probabilities represents.
c. Calculate and interpret P(A|C) and P(A/ ).
ANSWER:
a. P(A) = .13 + .10 + .11 + .11 =.45,
P(B) = .11 + .15 = .26
P(A B) = .11
P(A B) .11
b. P(A|B) = P(B) .26 .4231
Knowing that the car is black, the probability that it has an automatic transmission is .4231.
P(B|A) = P(A B) .11 .2444
P(A) .45
Knowing that the car has an automatic transmission, the probability that it is black is .2444.
P(A C) .13
c. P(A|C) = P(C) .28 .4643
The probability that the car has automatic transmission, knowing that the car is white is .4643.
P(A|C ) = P(A C ) .32 .4444
P(C ) .72
Knowing that the car is not white, the probability that it has an automatic transmission is .4444.
4. Consider the following information: where A = {Visa Card}, B = {MasterCard}, P(A) = .5, P(B) = .4, and
P(A B) = .25. Calculate each of the following probabilities.
a. P(B|A)
b. P(B |A)
c. P(A|B)
d. P(A |B)
e. Given that an individual is selected at random and that he or she has at least one card, what is the
probability that he or she has a Visa card?
ANSWER:
P(A B) .25
a. P(B|A) = P(A) .50 .50 b. P( B |A) = P(A B ) .25 .50
P(A) .50
P(A B) .25
c. P(A|B) = .6125
P(B) .40
d. P( A |B) = P(A B) .15 .3875
P(B) .40
P[A (A B)] .50
e. P(A|A B) = .7692
P(A B) .65
5. A certain shop repairs both audio and video components. Let A denote the event that the next component
brought in for repair is an audio component, and let B be the event that the next component is a compact
disc player (so the event B is contained in A). Suppose that P(A) = .625 and P(B) = .05. What is P(B/A)?
ANSWER:
Since B is contained in A, A B = B, then P(B|A) = P(A B) P(B) = .05 .08
P(A) P(A) .625
6. At a certain gas station, 40% of the customers use regular unleaded gas (), 35% use extra unleaded gas
1
(A2), and 25% use premium unleaded gas ( A3 ). Of those customers using regular gas, only 30% fill their
tanks (event B). Of those customers using extra gas, 60% fill their tanks, whereas of those using premium,
50% fill their tanks.
a. What is the probability that the next customer will request extra unleaded gas and fill the tank?
b. What is the probability that the next customer fills the tank?
c. If the next customer fills the tank, what is the probability that regular gas is requested? Extra gas?
Premium gas?
ANSWER:
P A .40, P A .35, P A .25
1 2 3
P B1/A .30
1
P B1/A .60
2
P B1/A 3 .50
Therefore,
P A B P A P B 1A .40 .30 .12
1 1 1
P A B P A P B 1 A .35 .60 .21
2 2 2
P A B P A P B 1A .25 .50 .125
3 3 3
a. P (A2 B) = .21
b. P(B) = P(A B) + P(A B) +P(A B) = .12+.21+.125=.455
1 2 3
P(A1 B) .12
c. P(P(A 1B) = .264
P(B) .455
P(A |B) = P(A 2 B) .21 .462

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