Topic 3: Probability IB Math Studies

Review Sheet for Probability

You should be able to do the following things on this test:

List the sample space of an event

Find the probability of a single event

Mathematically determine if two events are independent Find the probability of two or more dependent or independent events Draw and interpret Venn diagrams

Distinguish between inclusive and mutually exclusive events

Find the probability of inclusive and mutually exclusive events

Find the probability of conditional [“given that”] events

Use Venn diagrams, tree diagrams, tables, and lattice diagrams to solve problems Use the complement of an event to solve problems

11.

IB Math Studies

P.1 What is Probability?

Probability is the measure of how likely an event is to occur.

outcomes possible of number total

A to favorable outcomes of number

)(

)()event(

UnAn

AP

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]

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]

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? Because an event

can either occur or it can not occur, so P(A) + P(not A) = 1. Sometimes using the

complement is the easiest way to find the probability of “at least” events.

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?

H

T

1st

toss 2nd

toss 3rd

toss

Results

HHH

TTT

T C

3

U=25

P.1 In-Class Exercises

1. 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?

2. If two dice are rolled, what is the probability that both show the same

number?

3. If the probability of rain tomorrow is 52 , what is the probability of no rain?

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 Exercises

1. Black ace: 26

1

52

2 Not a black ace:

26

25

52

50

52

21

Diamond face card [KQJ]: 52

3 Jack or king:

13

2

52

8

2. Use a lattice diagram: 6

1

36

6

3. 5

3

5

21

4. Mr. or Mrs. Smith win: 214

1

4280

20

Any of the Smiths win: 535

4

4280

32

None of the Smiths win: 535

531

535

41

5. Sample space: AB AC AD BC BD CD

B and C = 6

1 Not C =

2

1

6

3

6.

Probability of neither = 210

29

1260

174

7. Divisible by 2 and 5: 10, 20. So, 10

1

20

2

8. There are 900 numbers. At least 1 zero: 900

171 no zeros:

900

729

900

1711

9. Nickel: 4

1

16

4 Not a penny:

16

9

10. Sample space: BBB, BBG, BGB, BGG, GBB, GGB, GBG, GGG

2 boys and 1 girl: 8

3 3 boys:

8

1

D C

518 454 114

174

U=1260

IB Math Studies

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 B)P(A . 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(B)P(A)B)P(A

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. A shelf contains 9 boxes of Corn Flakes and 6 boxes of Captain Crunch. Ahmed

chooses one box at random, then puts it back. A second person does the same

thing. What is the probability they both chose Captain Crunch? What is the

probability they both chose the same cereal?

3. 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.

4. It is given that 6.0)( AP , 7.0)( BP and 4.0)( BAP . Draw a Venn

diagram to represent this information. Are events A and B independent?

Why or why not?

5. It is given that 3

2)( AP and

6

1)( BAP . It is known that events A and B are

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:

A) following P(BP(A)B)P(A A) | P(BP(A)

6. 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?

7. 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.

P.2 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?

12. a) If 5

2)( AP and

3

2)( BP , find )( BAP if A and B are independent.

b) Given that 3

2)( AP ,

2

1)( BP , and

3

1)( BAP , determine whether A

and B are independent events.

P.2 Answers:

1. Independent: 309.08125

95

95

2. Dependent: 000495.0

3. a) Dependent: 0119.0841

71

82

93

b) Dependent: 070

81

92

4. Dependent: 0253.0

5. a) Dependent: 0476.0211

204

215

b) Independent: 0567.044125

215

215

6. a) Independent: 694.036

25

b) Independent: 833.06

5

7. Dependent: 0410.01958

137

144

154

8. Dependent: 246.0195

48

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 15

4B)P(A

b) They are independent ONLY IF P(B)P(A)B)P(A , so check…

IB Math Studies

P.3 Basic Set Theory

Shade each of these:

BA BA 'A

)'( BA )'( BA BA '

CB )( CBA ')( CBA

In a survey of children who saw three different shows at Walt Disney World, the

following information was gathered:

39 children liked The Little Mermaid

43 children liked 101 Dalmatians

56 children liked Mickey Mouse

7 children liked The Little Mermaid and 101 Dalmatians

10 children liked The Little Mermaid and Mickey Mouse

16 children liked 101 Dalmatians and Mickey Mouse

4 children liked The Little Mermaid, 101 Dalmatians, and Mickey Mouse

6 children did not like any of the shows

How many children were surveyed?

A B

C

A B

C

A B

C

A B A B A B

A B A B A B

Problems involving probability are often solved by using Venn Diagrams:

1. Given P(A) = 0.55, )( BAP = 0.7, and )( BAP = 0.2, find )'(BP

2. Given P(A) = 5

3, P(B) =

3

2, and )( BAP =

2

1, find )( BAP

3. Given that P(A) = 0.6, P(B) = 0.7, and that A and B are independent events, find

)( BAP , )'(AP , )'( BAP .

Hmmm… What does “independent” mean?

What formula did we learn?

A B

A B

A B

P.3 Exercises

1. 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?

2. 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?

3. 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?

4. 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.

5. 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?

6. Given P(C) = 0.44, )( DCP = 0.21 and )( DCP = 0.83, find P(D’).

7. If P(A) = 0.6 and P(B) = 0.5 and )( BAP = 0.2, find

a) )( BAP b) P(B’) c) )'( BAP

8. If P(A) = 0.6 and P(B) = 0.5 and A and B are independent events, find

a) )( BAP b) P(B’) c) )'( BAP

9. Given P(A) = 3

1 , P(B) = 8

3 , and B)P(A = 12

7 , show that A and B are

independent events.

P.3 Homework Answers

1. 25

10

2. 31

13

3. 56

20

4. 120

40

5. 18%

6. 0.40

7. a) 0.9 b) 0.5 c) 0.4

8. a) 0.8 b) 0.5 c) 0.3

9. Using the information, we can find that 24

3)( BAP . Since this is the same

as P(A)*P(B), these events are independent.

IB Math Studies

P.4 Combined and Mutually Exclusive Events: B)P(A

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.

B)P(AP(B)P(A)B)P(A

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 B

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 0 B)P(A

B)P(AP(B)P(A)B)P(A

P(B)P(A)B)P(A

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) = 10

3 and P(B) =

5

2,

find the value of B)P(A .

These two situations are easy to get confused. Be sure you know there are two

possible formulas for B)P(A :

Combined events: B)P(AP(B)P(A)B)P(A

Mutually exclusive events: P(B)P(A)B)P(A

A B

P.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) = 7

4 and P(Y) = 3

1

, find )( YXP and )( YXP .

5. Given P(S) = 0.34, and P(T) = 0.49, and )( TSP = 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 blonde-

haired person as homeroom representative to the student council?

P.4 Homework Answers:

1. Mutually exclusive: 769.013

10

2. Combined: 575.0200

115

3. Mutually exclusive and dependent: P(SS)+P(RR) =

304.0102

31

51

25

52

26

51

12

52

13

4. Mutually exclusive, so )( YXP = 0

Mutually exclusive, so )( YXP = 905.021

19

5. The events are mutually exclusive if 0)( TSP . So see if it works…

6. Combined: 0.11

7. a. 37

19 b.

37

12 c.

37

6

8. 6

5

9. 27

20

Try These

IB Math Studies

P.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, but I find it

confusing for solving real-life problems…

P(B)

B)P(AB)P(A

|

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(AB) = 0.6 and P(B) = 0.8.

a) P(A) b) P(A|B) c) P(A|B’)

P.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 3

1 )( BAP and

5

3)(BP , find 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.

8. Given 51)|( BAP and

21)( BP , find )( BAP .

9. Given 65)|( BAP ,

43)( AP , and

32)( BP , find )( BAP .

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

P.5 Homework Answers

1. a) 0.2625 b) 0.75 c) 0.4875 d) 0.712

2. 667.06

4

3. a) 32.025

8 b) 375.0

16

6

4. 556.09

5

5. a) 4

3 b) 1 c)

2

1 d)

10

3

6. 26.050

13 7. 833.0

6

5 8.

10

1

9. 861.036

31

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

11. 667.03

2

12. a) 88.025

22 b)

6

3

13. 551.049

27

14. a. 0.65 b. 0.35 c. 0.22 d. 0.28 e. 0.571 f. 0.25