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

Chapter 7

Probability

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

Definitions An experiment is a controlled operation that

yields a set of results. We don’t know the outcome in advance!

The possible results of an experiment are called its outcomes.

An event is a subcollection of the outcomes of an experiment.

Example – experiment: rolling a die. Outcomes: 1, 2, 3, 4, 5, 6 Some events: rolling an even number, rolling a 1

or a 6.

Slide 12 - 3Copyright © 2009 Pearson Education, Inc.

Definitions continued

Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment.

Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment.

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Empirical Probability

Example: In 100 tosses of a fair die, 19 landed showing a 3. Find the empirical probability of the die landing showing a 3.

Let E be the event of the die landing showing a 3.

P(E)

number of timesevent E has occurred

total number of times theexperiment has been performed

P(E)

19100

0.19

Slide 12 - 5Copyright © 2009 Pearson Education, Inc.

The Law of Large Numbers

The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals.

Slide 12 - 6Copyright © 2009 Pearson Education, Inc.

7.2 Theoretical Probability

Equally likely outcomes If each outcome of an experiment has the same

chance of occurring as any other outcome, they are said to be equally likely outcomes.

For equally likely outcomes, the probability of Event E may be calculated with the following formula.

P(E)

number of outcomes favorable to E total number of possible outcomes

Slide 12 - 7Copyright © 2009 Pearson Education, Inc.

Example

A die is rolled. Find the probability of rolling a) a 2. b) an odd number. c) a number less than 4. d) an 8. e) a number less than 9.

Slide 12 - 8Copyright © 2009 Pearson Education, Inc.

A die is rolled. Find the probability of rolling:

a) a 2.

b) an odd number. There are three ways an odd number can occur 1, 3 or 5.

c) a number less than 4. Three numbers are less than 4.

P(2)

number of outcomes that will result in a 2total number of possible outcomes

16

P(odd)

36

12

P(number less than 4)

36

12

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d) an 8. There are no outcomes that will result in an 8.

e) a number less than 9. All outcomes are less than 9. The event must occur and the probability is 1.

P(number greater than 8)

06

0

A die is rolled. Find the probability of rolling:

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Important Facts

The probability of an event that cannot occur is 0. The probability of an event that must occur is 1. Every probability is a number between 0 and 1

inclusive; that is, 0 <= P(E) <= 1. The sum of the probabilities of all possible

outcomes of an experiment is 1.

P

event happeningat least once

1 P

event doesnot happen

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Example

A standard deck of cards is well shuffled. Find the probability that the card is selected.a) a 10.b) not a 10.c) a card greater than 4 and less than 7.

Slide 12 - 13Copyright © 2009 Pearson Education, Inc.

Example continued

a) a 10There are four 10’s in a deck of 52 cards.

b) not a 10

P(10)

452

1

13

P(not a 10) 1 P(10)

11

13

1213

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Example continued

c) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s, and 6’s.

P > 4 and < 7 P 5 or 6

852

2

13

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7.4 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.

E P1 A1 P2 A2 P3 A3 ... Pn An

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Example

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

E 1

15075 149

150 3

75

150

447150

372150

2.48

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.

72

72

375

-2.5

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Fair Price

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 is 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

Fair price = expected value + cost to play

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Solution

$0

3/8

$10

$10$5-$8Amount Won/Lost

1/81/83/8Probability

$20$15$2Amt. Shown on Wheel

E 38

$8 38

$0 18

$5 18

$10

248

0 58

108

98

1.125 $1.13$10

$10

$2

$2

$20$15

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Solution

Fair price = expectation + cost to play = -$1.13 + $10

= $8.87

Thus, the fair price is about $8.87.

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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.

Tree diagrams are helpful for visualizing the different outcomes.

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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 possible outcomes.

b) Construct a tree diagramand list the outcomes.

a) 3 • 2 = 6 ways b)

B

P

B

G

BGPGP

PBPGBP

BGGP

GB

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Example Two balls are to be selected without replacement from a bag that

contains one purple, one blue, and one green ball.

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.

c)

d)

P blue 4

6

23

P Purple,Green P P,G 1

6 B

P

B

G

BGPGP

PBPGBP

BGGP

GB

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Example Two balls are to be selected without replacement from a bag that

contains one purple, one blue, and one green ball.

e) Find the probability that a blue ball is NOT selected.

e) P blue 4

6

23

B

P

B

G

BGPGP

PBPGBP

BGGP

GB

2 1no blue6 3

P

P

event happeningat least once

1 P

event doesnot happen

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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

2652

2652

12

12

14

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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 2552 511 25 252 51 102

P A P B P P

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Or Problems

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

One card is selected from a standard deck of playing cards. Determine the probability of the following events.a) selecting a 3 or a jackb) selecting a jack or a heartc) selecting a picture card or a red cardd) selecting a red card or a black card

Slide 12 - 27Copyright © 2009 Pearson Education, Inc.

Solutions

a) 3 or a jack

b) jack or a heart

P 3 P jack 452

452

852

2

13

P jack P heart Pjack andheart

452

1352

1

52

1652

4

13

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Solutions continued

c) picture card or red card

d) red card or black card

P picture P red Ppicture &red card

1252

2652

652

3252

8

13

P red P black 2652

2652

5252

1

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Odds in Favor

Odds in favor of event P event occurs

P event fails to occur

P success P failure

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Example

Find the odds in favor of landing on blue in one spin of the spinner.

The odds in favor of spinning blue are 3:5.

P blue 3

8 P not blue 5

8

odds in favor

3858

38

85

35

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7.3 Odds

Odds Against

Odds against event P event fails to occur

P event occurs

P failure P success

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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.

Slide 12 - 33Copyright © 2009 Pearson Education, Inc.

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.

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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

630

529

428

1

203

Slide 12 - 35Copyright © 2009 Pearson Education, Inc.

Solution

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

1430

1029

1328

13174

Slide 12 - 36Copyright © 2009 Pearson Education, Inc.

Solution

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.

Pnoneyellow

P

first notyellow

P

second notyellow

P

third notyellow

2030

1929

1828

57203

Slide 12 - 37Copyright © 2009 Pearson Education, Inc.

Solution

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)

157203

146203