Chapter 8: Sets and Probability
Day 1:
Day 2:
Day 3:
Day 4:
Day 5:
Day 6:
Day 7:
SP.1 Sets
SP.2 Intro to Probability
SP.3 Independent and Dependent Events + Venn
Diagrams/Tree Diagrams
Quiz
SP.4 Combined and Mutually Exclusive Events
SP.5 Conditional Probability
Quiz
Review
Test on Sets and Probability
P.1 Intro to Probability
Probability is the measure of how likely an event is to occur.
P (event A) 
n (A) number of outcomes favorable to A

n (U ) total number of possible outcomes
A is the event you want to occur.
U is the universal set of all possibilities.
1. A spinner has 4 equal sectors colored yellow, blue, green and red.
What are the chances of landing on blue after spinning the
spinner?
2. A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single
marble is chosen at random from the jar, what is the probability
of choosing a red marble? a green or a yellow marble?
3. A teacher chooses a student at random from a class of 30 girls. What is the
probability that the teacher chooses a girl? a boy?
A “sample space” is a listing of all possible outcomes of an event. There are three
common ways to show a sample space: lattice, tree, and Venn diagrams.
4. A red die and a white die are thrown. What is the probability that the sum of
the numbers showing on the dice is 9 or 10? [lattice diagram]
Red
White
1
2
3
4
5
6
1
2
3
4
5
6
5. Suppose you toss a coin three times. What is the probability that exactly two
of the tosses results in “heads”? [tree diagram]
1st
toss
2nd
toss
Results
3rd
toss
HHH
H
T
TTT
6. In a class of 25 students, it is found that 16 of the students play tennis, 6 play
both tennis and chess, and 3 do not participate in any activities at all. Find the
probability that a student plays chess. [Venn diagram]
T
C
U=25
3
The probability of an event NOT occurring is called the “complement” of the
event. It is written P(A’). We might say that P(not A) = 1 – P(A). Why?
7. A collection of 38 computer disks contain four that are defective. One is
chosen at random. What is the probability it’s defective? it’s not defective?
8. Of seventeen students in a class, five have blue eyes. One student is chosen at
random. What is the probability the student has blue eyes? doesn’t have blue
eyes?
SP.2
1.
2.
3.
Exercises
A card is chosen from a well-shuffled deck. What is the probability that it is a
black ace? not a black ace? a diamond face card? a jack or a king?
If two dice are rolled, what is the probability that both show the same
number?
If the probability of rain tomorrow is 2 , what is the probability of no rain?
5
4.
Mr. & Mrs. Smith each bought 10 raffle tickets, and each of their three
children bought 4 tickets. If 4280 tickets were sold in all, what is the
probability that the grand prize winner is Mr. or Mrs. Smith? one of the 5
Smiths? none of the Smiths?
5.
From a group consisting of Amal, Bara, Cesar, and Denay, two people are to
be randomly selected to serve on a committee. List the sample space for this
experiment. Find the probability that Bara and Cesar are selected. Find the
probability that Cesar is not selected.
6.
Of the 1260 households in a small town, 632 have dogs, 568 have cats, and
114 have both types of pets. Construct a Venn Diagram. If a household is
chosen at random, what is the probability that the household has neither a
cat nor a dog?
7.
A bag has 20 discs numbered from 1 to 20. A disc is drawn at random . What
is the probability that the disc has a number that is divisible by 2 and 5?
8.
A number between 100 and 999, inclusive, is chosen at random. What is the
probability that it contains at least one zero? no zeros?
9.
A bag contains 7 pennies, 4 nickels, and 5 dimes. What is the probability of
choosing a nickel? of not choosing a penny?
10. A family has 3 children. List the sample space, assuming birth order matters.
What the probability that there are 2 boys and 1 girl? that there are 3 boys?
P.1 Answers to In-Class Exercises
1.
Black ace:
2
1

52 26
2
50 25


52 52 26
8
2
Jack or king:

52 13
Not a black ace: 1 
3
52
6
1
Use a lattice diagram:

36 6
2 3
1 
5 5
20
1

Mr. or Mrs. Smith win:
4280 214
32
4
Any of the Smiths win:

4280 535
4
531

None of the Smiths win: 1 
535 535
Diamond face card [KQJ]:
2.
3.
4.
5.
Sample space:
B and C =
6.
AB AC AD BC BD CD
1
6
Not C =
D
3 1

6 2
U=1260
C
518
114
454
174
Probability of neither =
174
29

1260 210
2
1

20 10
7.
Divisible by 2 and 5: 10, 20. So,
8.
There are 900 numbers. At least 1 zero:
9. Nickel:
4
1

16 4
Not a penny:
171
900
no zeros: 1 
9
16
10. Sample space: BBB, BBG, BGB, BGG, GBB, GGB, GBG, GGG
2 boys and 1 girl:
3
8
3 boys:
1
8
171 729

900 900
SP.2 Exercises
1.
2.
Which of the following numbers cannot be the probability of some event?
Explain your reasoning for each.
0.71
4.1
1
8
- 0.5
1.21
0.5
0
1
150%
13
5
Isabel Briggs Myers was a pioneer in the study of personality types. Do
married couples choose similar or different personality types in their
mates? The following data give an indication.
Number of Similar Preferences
All 4 type preferences
Three type preferences
Two type preferences
One type preference
None
Number of Married Couples
34
131
124
71
15
Suppose that a married couple is selected at random.
a) What is the probability that they will have no preferences in common?
One preference? 2 preferences? 3 preferences? 4 preferences?
b) What is the sample space? What do the probabilities add up to? Does
this make sense?
3.
4.
A botanist has developed a new hybrid cotton plant that can withstand
insects better than other cotton plants. However, there is some concern
about the germination of seeds from the new plant. A random sample of
3000 seeds planted in warm, moist soil showed 2430 to germinate.
a)
What is the probability that a seed will germinate?
b)
What is the complement of this event? What is its probability?
When do creative people get their best ideas? USA Today did a survey of
inventors who hold U.S. patents.
Time
Number of Inventors
6 a.m – 12 noon
290
12 noon – 6 p.m.
135
6 p.m. – 12 midnight
319
12 midnight – 6 a.m.
222
a)
b)
What is the probability that an inventor has a “best idea” from 12
noon to 12 midnight? from 12 midnight to 12 noon?
What is the sample space?
5.
About 56% of the general population wears corrective eyeglasses, while
3.6% wears contact lenses. [Assume that no one wears both at once…]
a)
b)
c)
6.
What ‘events’ form the sample space? What are their probabilities?
Make sure they add up to one…
What idea of the ‘complement’ do you see in this problem?
Draw a Venn diagram that represents this information
Xavier runs a computer store. Yesterday, 127 people walked by his store.
Of those, 58 came into the store. Of those 58, 25 bought something.
a)
b)
c)
d)
e)
Draw a Venn diagram to represent this sample space.
What is the probability that a person who walks by the store will
enter the store?
What is the probability that a person who enters the store will buy
something?
What is the probability that a person who walks by the store will buy
something?
What is the probability that a person who comes into the store will
buy nothing?
P.2 Independent and Dependent Events
Today we will look at the probability of two events both happening. The notation
for this is P(A and B) or P(A  B) . Tree diagrams are often the easiest way to
organize these kinds of events.
Two events are independent if the outcome of the first event does not influence
the outcome of the second event. The following formula applies:
P(A  B)  P(A)  P(B)
1. Find the probability of getting a sum of 5 on the first toss of two dice and a
sum of 3 on the second toss.
2. Andrew is 55, and the probability that he will be alive in 10 years is 0.72. Ellen
is 35, and the probability that she will be alive in 10 years is 0.92. What is the
probability that a) they will both be alive in 10 years; b) neither will be alive in
10 years; c) one of them will be alive in 10 years.
3. It is given that P (A)  0.6 , P (B )  0.7 and P (A  B )  0.4 . Draw a Venn
diagram to represent this information. Are events A and B independent?
Why or why not?
4. It is given that P (A ) 
2
1
and P (A  B )  . It is known that events A and B are
6
3
independent. Find P(B).
Two events are dependent if the outcome of the first event in some way
influences the outcome of the second event. The following formula applies:
P(A  B)  P(A)  P(B given A)  P(A)  P(B | A)
5. There are 2 cans of root beer and 4 cans of Dr. Pepper on the counter. Nada
drinks two of them at random. What is the probability that she drank one can
of each?
6. A quality-control procedure for testing Ready-Flash disposable cameras
consists of choosing two cameras at random from each lot of 100 without
replacement. If both cameras are defective, the entire lot is rejected.
Typically, 10 cameras of the 100 are defective. Find the probability that the lot
of cameras will NOT be rejected.
Probability + Set Theory
Problems involving probability are often solved by using Venn Diagrams:
1. Given P(A) = 0.55, P (A  B ) = 0.7, and P (A  B ) = 0.2, find P (B ' )
A
B
2. Given P(A) =
3
2
1
, P(B) = , and P (A  B ) = , find P (A  B )
5
3
2
A
B
3. Given that P(A) = 0.6, P(B) = 0.7, and that A and B are independent events, find
P (A  B ) , P (A') , P (A 'B ) .
A
Hmmm… What does “independent” mean?
What formula did we learn?
B
SP.3 Homework Exercises
The more complicated the problem, the easier it will be to solve if you use a TREE
diagram…
1. A bowl contains 5 oranges and 4 tangerines. Marie randomly selects one,
puts it back, and then selects another. What is the probability that both
selections were oranges?
2. Find the probability of being dealt 5 hearts from a standard deck of 52 cards.
3. Badriya’s wallet contains four 1BD notes, three 5BD notes, and two 10BD
notes. a) Find the probability of selecting three 5BD notes. b) Find the
probability of selecting three 10BD notes.
4. For a bingo game, wooden balls numbered consecutively from 1 to 75 are
placed in a box. Five balls are drawn at random without replacement. Find
the probability of selecting five even numbers.
5. There are 5 pennies, 7 nickels, and 9 dimes in Venika’s coin collection. She
chooses two coins at random from the collection. What is the probability
that both are pennies if a) no replacement occurs; b) replacement occurs?
6. Two dice are tossed. a) Find P(no 2s). b) Find P(two different numbers).
7. A bag contains 4 red, 4 green, and 7 blue marbles. Three are selected in
sequence without replacement. What is the probability of selecting a red, a
green, and a blue marble in that order?
8. A bag contains 4 red, 4 green, and 7 blue marbles. Three are selected in
sequence without replacement. What is the probability of selecting a red, a
green, and a blue marble in any order?
9. There are two traffic lights along the route that Aly rides home from school.
one traffic light is red 50% of the time. The next traffic light is red 60% of the
time. a) What is the probability that Aly will hit both green lights on the way
home? b) What is the probability that he will hit one green light on the way
home?
10. A student runs the 100m, 200m, and 400m races at the school athletics day.
She has an 80% chance of winning any given race. a) Find the probability
that she will win all 3 races. b) Find the probability that she will win any 2
races.
11. Dale and Kritt are trying to solve a physics problem. Dale has a 65% chance
of solving the problem, while Kritt has a 75% chance. a) What is the
probability that the problem gets solved? b) What is the probability that the
problem does not get solved?
2
2
and P (B )  , find P (A  B ) if A and B are independent.
5
3
2
1
1
b) Given that P (A )  , P (B )  , and P (A  B )  , determine whether A
3
2
3
12. a) If P (A) 
and B are independent events.
13. A platform diving squad of 25 members has 18 members who dive from 10 m
and 17 who dive from 4 m. What is the probability that a member of the
squad dives from both platforms?
14. A badminton club has 31 playing members. 28 play singles and 16 play
doubles. What is the probability that a member plays both singles and
doubles?
15. In a factory, 56 people work on the assembly line. 47 work day shifts and 29
work night shifts. What is the probability that an employee works both day
shifts and night shifts?
16. In a group of 120 students, 75 know how to use a Macintosh, 65 know how to
use a PC, and 20 do not know how to use either. Find the probability that a
student knows how to use both kinds of computers.
17. A city has three newspapers A, B and C. Of the adult population, 1% read none
of these newspapers, 36% read A, 40% read B, 52% read C, 8% read A and B,
11% read B and C, 13% read A and C and 3% read all three papers. What
percentage of the adult population read newspaper A only?
18. Given P(C) = 0.44, P (C D) = 0.21 and P (C D) = 0.83, find P(D’).
19. If P(A) = 0.6 and P(B) = 0.5 and P (A  B ) = 0.2, find
a)
b)
P(B’)
c)
P (A  B )
P (A  B')
20. If P(A) = 0.6 and P(B) = 0.5 and A and B are independent events, find
a)
b)
P(B’)
c)
P (A  B )
P (A  B')
21. Given P(A) =
1
,
3
P(B) =
independent events.
3
,
8
and P(A B) =
7
12
, show that A and B are
SP.3 Homework Answers:
1.
Independent:
5 5 25
 
 0.309
9 9 81
2.
Dependent:
0.000495
3.
a) Dependent:
b) Dependent:
4.
Dependent:
5.
a) Dependent:
b) Independent:
6.
a) Independent:
b) Independent:
7.
Dependent:
8.
Dependent:
3 2 1
1
  
 0.0119
9 8 7 84
2 1 0
  0
9 8 7
0.0253
5 4
1


 0.0476
21 20 21
5 5
25
 
 0.0567
21 21 441
25
 0.694
36
5
 0.833
6
4 4 7
8
  
 0.0410
15 14 13 195
48
 0.246
195
9.
a) Independent:
P(GG) = 0.50 * 0.40 = 0.20
b) Independent:
P(RG  GR) = (0.50 * 0.40) + (0.50 * 0.60) = 0.50
10. Independent: a) 0.512
b) 0.384
11. Independent: From a tree diagram:
a) P(it gets solved) = 1-P(doesn’t get solved) = 0.9125
b) P(doesn’t get solved) = 0.35*0.25 = 0.0875
12. a) Independent, so P(A  B) 
4
15
b) They are independent ONLY IF P(A  B)  P(A)  P(B) , so check…
13. 10
18.0.40
14. 13
19.a) 0.9
b)0.5
c) 0.4
20.a) 0.8
b)0.5
c) 0.3
25
31
15. 20
56
16. 40
120
17.18%
P.3: Expected Value
21.Using the information, we can find
that P (A  B ) 
3
. Since this is the
24
same as P(A)*P(B), these events are
independent.
P.4 Combined and Mutually Exclusive Events: P(A  B)
Two either/or events are combined if there exists the possibility that both events
might occur at the same time. We have to consider the possibility of A occurring
by itself, B occurring by itself, and both A and B occurring together. Because
there is an area of overlap, we have to avoid double counting by subtracting the
area of overlap.
A
B
P(A  B)  P(A)  P(B)  P(A  B)
1. When choosing a card, what is the possibility of choosing a king or a diamond?
2. Professor Jackson is in charge of a program to prepare people for a high school
equivalency exam. Records show that 80% of the students need work in math,
70% need work in English, and 55% need work in both areas. Compute the
probability that a student selected at random needs help in math or English.
3. In a bag are 100 discs numbered 1 to 100. A disc is selected at random from
the bag. Find the probability that the number on the selected disc is even or a
multiple of 5.
4. A garage knows that when a person calls to report that their car won’t start,
the probability that the engine is flooded is 0.5 and the probability that the
battery is dead is 0.4, and the probability of both is 0.1. What is the
probability that the next person who calls will have either a flooded engine or
a dead battery? Are these events dependent or independent? Why?
5. In a class, half the pupils study Mathematics, a third study English, and a
quarter study both Mathematics and English. Find the probability that a
student selected at random studies either Mathematics or English.
A special case of combined events occurs if either event may occur, but both
cannot happen at the same time. We call these mutually exclusive events. In this
case, there is no overlap, so we don’t have to worry about double counting. It’s
the same equation as the one we saw on the previous page, but with P(A  B)  0
A
B
P(A  B)  P(A)  P(B)  P(A  B)
P(A  B)  P(A)  P(B)
6. When tossing a die, what is the probability of tossing a 3 or a 4?
7. When choosing a card, what is the probability of choosing a jack or a king?
8. The Cost Less Clothing Store carries “seconds” in slacks that don’t quite fit. If
you buy a pair of slacks in your regular waist size without trying them on, the
probability that the waist will be too tight is 0.30 and the probability that it will
be too loose is 0.10. What is the probability that the waist won’t fit?
9. Given that events A and B are mutually exclusive with P(A) =
find the value of P(A B) .
3
2
and P(B) = ,
5
10
These two situations are easy to get confused. Be sure you know there are two
possible formulas for P(A B) :
Combined events:
Mutually exclusive events:
P(A  B)  P(A)  P(B)  P(A  B)
P(A  B)  P(A)  P(B)
SP.4 Homework Exercises:
For each problem, tell whether the event is combined or mutually exclusive. Then
solve the problem using the appropriate technique, formula, or diagram.
1. Lara has 4 pennies, 3 nickels, and 6 dimes in her pocket. She takes one coin
from her pocket at random. What is the probability that it is a penny or a
dime?
2. After a recent disaster, 200 people in a community were asked what kind of
help they gave to the victims. 65 said they donated food. 50 people said they
donated money, and 30 people said they donated both. What is the
probability that a person selected at random from the sample donated neither
food nor money?
3. Two cards are chosen from a standard deck of 52 cards. What is the
probability that both are spades or both are red cards?
4. Given that the events X and Y are mutually exclusive with P(X) =
1
,
3
4
7
and P(Y) =
find P (X Y ) and P (X Y ) .
5. Given P(S) = 0.34, and P(T) = 0.49, and P (S T ) = 0.83, show that the events S
and T are mutually exclusive.
6. As a result of a survey of the households in a town, it is found that 80% have a
video recorder and 24% have satellite television. Given that 15% have both a
video recorder and satellite television, find the proportion of households with
neither a video recorder nor satellite television.
7. When a roulette wheel is spun, the score will be a number from 0 through 36.
Each score is equally likely. Find the probability that the score is
a. an even number
b. a multiple of 3
c. a multiple of 6
8. From a group consisting of Alvin, Bob, Carol and Donna, two people are to be
randomly selected to serve on a committee. Use a tree diagram to give the
sample space. What is the probability that Bob or Carol is selected?
9. In a homeroom, 5 of the 12 girls have blonde hair and 6 of the 15 boys have
blonde hair. What is the probability of randomly selecting a boy or a blondehaired person as homeroom representative to the student council?
SP.4 Homework Answers:
1. Mutually exclusive:
2. Combined:
10
 0.769
13
115
 0.575
200
3. Mutually exclusive and dependent:
 13 12   26 25  31
 

 0.304


 52 51   52 51  102
P(SS)+P(RR) =
4. Mutually exclusive, so P (X Y ) = 0
Mutually exclusive, so P (X Y ) =
19
 0.905
21
5. The events are mutually exclusive if P (S T )  0 . So see if it works…
6. Combined:
7.
a.
8.
5
6
9.
20
27
19
37
0.11
b.
12
37
c.
6
37
SP.5 Conditional Probability
Conditional probability measures the probability of an event A occurring given
that B has already occurred. There is a formula for this in your packets.
Sometimes it can be confusing for solving real-life problems…
P(A |B) 
P(A  B)
P(B)
In conditional probability, what you are doing is reducing the sample space to B
and then finding how much A there is in B. Venn, tree, and lattice diagrams can
help…
1. Two dice numbered one to six are rolled. Find the probability of obtaining a
sum of five given that the sum is seven or less. P(sum of 5|sum of 7 or less)
2. In a class of 25 students it is found that 5 of the students play both tennis and
chess, 10 play tennis only, and 3 do not participate in any activities. Find the
probability that a student selected at random from this group plays tennis,
given the student plays chess.
3. Bag A contains 5 blue and 4 green marbles. Bag B contains 3 yellow, 4 blue,
and 2 green marbles. Given you have a green marble, what is the probability it
came from Bag A?
4. At the basketball game, Amanda got into a two-shot foul situation. She figured
her chance of making the first shot was 0.7. If she made the first shot, her
chance of making the second shot was 0.6. However, if she missed the first
shot, her probability of making the second shot was only 0.4. Given Amanda
missed the second shot, find the probability that she made the first shot.
5. The events A and B are independent. P(AB) = 0.6 and P(B) = 0.8.
a)
P(A)
b)
P(A|B)
c)
P(A|B’)
SP.5 Homework Exercises
1.
Dana and Lana are trying to solve a physics problem. Dana has a 65%
chance of solving the problem, and Lana has a 75% chance. Find the
probability that
a. only Lana solves the problem.
b. Lana solves the problem.
c. both solve the problem.
d. Dana solves the problem, given the problem was solved.
2.
What is the probability that the total of two dice will be greater than 8
given that the first die is a 6?
3.
In a class of 25 students, 14 like pizza and 16 like coffee. One student likes
neither. One student is randomly selected from the class. What is the
probability that the student:
a) likes pizza, but not coffee
b) likes pizza given that s/he likes coffee?
4.
Given P (A  B ) 
5.
A drawer contains three good light bulbs and two defective light bulbs.
Two light bulbs are chosen at random without replacement. Find each
probability:
a. P(2nd good | 1st defective)
b. P(good  defective)
c. P(2nd good | 1st good)
d. P(good  good)
6.
Three cards are drawn from a standard deck of 52 cards. What is the
probability that the third card is a spade if the first two cards are hearts?
7.
400 families were surveyed. It was found that 90% had a TV set and 60%
had a computer. Every family had at least one of these items. If one of
these families is randomly selected, find the probability it has a TV set given
that it has a computer.
3
1
and P (B )  , find P(A|B).
5
3
8.
Given P (A | B )  1 and P (B )  1 , find P (A  B ) .
9.
Given P (A | B )  5 , P (A)  3 , and P (B )  2 , find P (A  B ) .
2
5
6
4
3
11.
Urn 1 contains 4 red and 6 green balls while Urn 2 contains 7 red and 3
green balls. An urn is chosen at random and then a ball is chosen from the
selected urn. Draw a tree diagram. Find P(Urn 1|G).
12.
Thirty students sit for an examination in both French and English. 25 pass
French, 24 pass English, and 3 fail both. Determine the probability that a
student who
a. passed French also passed English
b. failed English passed in French
13.
The probability that a animal will still be alive in 12 years is 0.55. The
probability that its mate will still be alive in 12 years is 0.60. Find the
probability that the mate is still alive in 12 years given that only one is still
alive
14.
In a certain town, 3 newspapers are published. 20% of the population read
A, 16% read B, 14% read C, 8% read A and B, 5% read A and C, 4% read B
and C, and 2% read all newspapers. A person is selected at random.
Determine the probability that the person reads:
a. none of the papers
b. at least one of the papers
c. exactly one of the papers
d. either A or B
e. A, given that the person reads at least one paper
f. C, given that the person reads either A or B or both
SP.5 Homework Answers
1. a)
2.
b)
0.75
c)
0.4875
d)
0.712
c)
1
2
d)
3
10
4
 0.667
6
3. a)
4.
0.2625
8
 0.32 b)
25
6
 0.375
16
5
 0.556
9
5. a)
3
4
6.
13
 0.26
50
9.
31
 0.861
36
b)
1
7.
5
 0.833
6
8.
1
10
10.a) 0.9568
b) 0.95
c) 0.99
d) it would be great if both were higher -- if 100% of innocent people were
acquitted and 100% of guilty people were convicted
e) 0.7902
f) 0.1978
2
3
11.  0.667
12.a)
13.
22
 0.88
25
b)
3
6
27
 0.551
49
14.a. 0.65
b. 0.35
c. 0.22
d. 0.28
e. 0.571
f. 0.25