Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 1 Unit 7 Probability

Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 1

Unit 7

Probability

Chapter 12 Section 1 - Slide 2Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN• Empirical probability and theoretical

probability• Compound probability, conditional

probability, and binomial probability• Odds against an event and odds in

favor of an event• Expected value• Tree diagrams


WHAT YOU WILL LEARN

• Mutually exclusive events and independent events

• The counting principle, permutations, and combinations


Section 4

Expected Value (Expectation)


Expected Value

The symbol P1 represents the probability that the first event will occur, and A1 represents the net amount won or lost if the first event occurs.

P2 is the probability of the second event, and A2

is the net amount won or lost if the second event occurs.

And so on…

E P1A

1 P

2A

2 P

3A

3 ... P

nA

n


Example

Teresa is taking a multiple-choice test in which there are four possible answers for each question. The instructor indicated that she will be awarded 3 points for each correct answer and she will lose 1 point for each incorrect answer and no points will be awarded or subtracted for answers left blank. If Teresa does not know the correct answer to

a question, is it to her advantage or disadvantage to guess?

If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at the answer?


Solution Expected value if Teresa guesses.

Therefore, over the long run, Theresa will neither gain nor lose points by guessing.

P guesses correctly 1

4

E 1

43 3

4 1

3

4

3

40

P guesses incorrectly 3

4


Solution (continued) —eliminate a choice

Therefore, over the long run, Theresa will, on average, gain 1/3 point each time she guesses when she can eliminate one choice.

P guesses correctly 1

3

E 1

33 2

3 1

12

3

1

3

P guesses incorrectly 2

3


Example: Winning a Prize

When Calvin Winters attends a tree farm event, he is given a free ticket for the $75 door prize. A total of 150 tickets will be given out. Determine his expectation of winning the door prize.

Therefore, Calvin’s expectation is $0.50, or 50 cents.

E 1

15075 149

1500

1

2


Example

When Calvin Winters attends a tree farm event, he is given the opportunity to purchase a ticket for the $75 door prize. The cost of the ticket is $3, and 150 tickets will be sold. Determine Calvin’s expectation if he purchases one ticket.


Solution

Calvin’s expectation is $2.50 when he purchases one ticket.

1 14972 3

150 15072 447

150 150375

1502.50

E

Slide 12 - 12Copyright © 2009 Pearson Education, Inc.

You get to select one card at random from a standard deck of 52 cards. If you pick a king, you win $6. If you pick a queen, you lose $3 and if you pick a jack, you lose $2. Determine your expectation for this game.

a. $0.08

b. $0.46

c. $0.77

d. $1.00


You get to select one card at random from a standard deck of 52 cards. If you pick a king, you win $6. If you pick a queen, you lose $3 and if you pick a jack, you lose $2. Determine your expectation for this game.

a. $0.08

b. $0.46

c. $0.77

d. $1.00


Fair Price

Fair price = expected value + cost to play


Example

Suppose you are playing a game in which you spin the pointer shown in the figure, and you are awarded the amount shown under the pointer. If it costs $10 to play the game, determine:

a) the expectation of the person who plays the game.

b) the fair price to play the game.

$10

$10

$2

$2

$20$15


Solution

$0

3/8

$10

$10$5$8Amount Won/Lost

1/81/83/8Probability

$20$15$2Amt. Shown on Wheel

E 3

8 $8 3

8$0 1

8$5 1

8$10

24

8 0

5

8

10

8

9

8 1.125 $1.13


Solution

Fair price = expectation + cost to play

= $1.13 + $10

= $8.87

Thus, the fair price is about $8.87.


Section 5

Tree Diagrams


Counting Principle

If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M • N distinct ways.


Definitions

Sample space: A list of all possible outcomes of an experiment.

Sample point: Each individual outcome in the sample space.

Tree diagrams are helpful in determining sample spaces.


Example

Two balls are to be selected without replacement from a bag that contains one purple, one blue, and one green ball.

a) Use the counting principle to determine the number of points in the sample space.

b) Construct a tree diagram and list the sample space.

c) Find the probability that one blue ball is selected.

d) Find the probability that a purple ball followed by a green ball is selected.


Solutions

a) 3 • 2 = 6 ways

b)

c)

d)

B

P

B

G

B

G

P

G

P

PB

PGBP

BGGP

GB

P blue 4

6

2

3

P Purple,Green P P,G 1

6


One die is rolled and one colored chip - black or white - is selected at random. Use the counting principle to determine the number of sample points in the sample space.

a. 6

c. 12

b. 8

d. 10


One die is rolled and one colored chip - black or white - is selected at random. Use the counting principle to determine the number of sample points in the sample space.

a. 6

c. 12

b. 8

d. 10


P(event happening at least once)

P

event happening

at least once

1 P

event does

not happen


Section 6

Or and And Problems


Or Problems

P(A or B) = P(A) + P(B) P(A and B) Example: Each of the numbers 1, 2, 3, 4, 5, 6,

7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains an even number or a number greater than 5.


Solution

P(A or B) = P(A) + P(B) P(A and B)

Thus, the probability of selecting an even number or a number greater than 5 is 7/10.

Peven or

greater than 5

P even P greater than 5 Peven and

greater than 5

5

10

5

10

3

10

7

10


Example

Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains a number less than 3 or a number greater than 7.


Solution

There are no numbers that are both less than 3 and greater than 7. Therefore,

P less than 3 2

10

P greater than 7 3

10

P

less than 3 or

greater than 7

2

10

3

10 0

5

10

1

2


Mutually Exclusive

Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously.


Example

One card is selected from a standard deck of playing cards. Determine the probability of the following events.

a) selecting a 3 or a jack

b) selecting a jack or a heart

c) selecting a picture card or a red card

d) selecting a red card or a black card


Solutions

a) 3 or a jack (mutually exclusive)

b) jack or a heart

P 3 P jack 4

52

4

52

8

52

2

13

P jack P heart Pjack and

heart

4

52

13

52

1

52

16

52

4

13


Solutions continued

c) picture card or red card

d) red card or black card (mutually exclusive)

P picture P red Ppicture &

red card

12

52

26

52

6

52

32

52

8

13

P red P black 26

52

26

52

52

521


And Problems

P(A and B) = P(A) • P(B) Example: Two cards are to be selected with

replacement from a deck of cards. Find the probability that two red cards will be selected.

P A P B P red P red

26

5226

52

1

21

2

1

4


Example

Two cards are to be selected without replacement from a deck of cards. Find the probability that two red cards will be selected.

red red

26 25

52 511 25 25

2 51 102

P A P B P P


Independent Events

Event A and Event B are independent events if the occurrence of either event in no way affects the probability of the occurrence of the other event.

Experiments done with replacement will result in independent events, and those done without replacement will result in dependent events.


One die is rolled and one colored chip - black or white - is selected at random. Determine the probability of obtaining an even number and the color white.

a.

c.

1

2

1

3b.

d.

1

6

1

4


One die is rolled and one colored chip - black or white - is selected at random. Determine the probability of obtaining an even number and the color white.

a.

c.

1

2

1

3b.

d.

1

6

1

4


Example

A package of 30 tulip bulbs contains 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. Three bulbs are randomly selected and planted. Find the probability of each of the following.a.All three bulbs will produce pink flowers.b.The first bulb selected will produce a red

flower, the second will produce a yellow flower and the third will produce a red flower.

c.None of the bulbs will produce a yellow flower.

d.At least one will produce yellow flowers.


Solution

30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers.

a. All three bulbs will produce pink flowers.

P 3 pink P pink 1 P pink 2 P pink 3

6

30

5

29

4

28

1

203


Solution (continued)

30 tulip bulbs, 14 bulbs for red flowers,0010 for yellow flowers, and 6 for pink flowers.

b. The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower.

P red, yellow, red P red P yellow P red

14

3010

2913

28

13

174



30 tulip bulbs, 14 bulbs for red flowers,0010 for yellow flowers, and 6 for pink flowers.

c. None of the bulbs will produce a yellow flower.

Pnone

yellow

P

first not

yellow

P

second not

yellow

P

third not

yellow

20

3019

2918

28

57

203



30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers.

d. At least one will produce yellow flowers.

P(at least one yellow) = 1 P(no yellow)

157

203

146

203

Documents

Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 1 Unit 7 Probability