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

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

AND

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

Unit 7

Probability


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 1

The Nature of Probability


Definitions

An experiment is a controlled operation that yields a set of results.

The possible results of an experiment are called its outcomes.

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


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.


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

9Copyright © 2009 Pearson Education, Inc.

Of 12 children playing at the playground, 4 are playing on the swing set. Determine the empirical probability that the next child to the playground will play on the swing set.

a.

c.

12

13

b.

d. 14

16


Of 12 children playing at the playground, 4 are playing on the swing set. Determine the empirical probability that the next child to the playground will play on the swing set.

a.

c.

12

13

b.

d. 14

16


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.


Section 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


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.


Solutions: There are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. a)

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

c) 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


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

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

Solutions: There are six equally likely outcomes: 1, 2, 3, 4, 5, and 6 (continued)

P(number greater than 8)

06

0


Each of the numbers 0-9 is written on a sheet of paper and the ten sheets of paper are placed in a box. If one sheet of paper is selected at random from the box, determine the probability that the number selected is less than 3.

a.

c.

12

13

b.

d. 3

10 7

10


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.


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 heart.d) an ace, 2, or 3.e) diamond and spade.f) a card greater than 4 and less than 7.


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


Example continued

c) a heart

There are 13 hearts in the deck.

d) an ace, 2 or 3

There are 4 aces, 4 twos and 4 threes, or a total of 12 cards.

P(heart)

1352

14 = =12 3

(A, 2, or 3)52 13

P


Example continued

d) diamond and spadeThe word and means both events must occur. This is not possible.

e) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s and 6’s (or a total of 8 cards).

P > 4 and < 7 P 5 or 6

852

2

13

P(diamond and spade)

052

0


Section 3

Odds


Odds Against

Odds against event P event fails to occur

P event occurs

P failure P success


Example: Odds Against

Example: Find the odds against rolling a 5 on one roll of a die.

The odds against rolling a 5 are 5:1.

P 5 1

6 P fails to roll a 5 5

6

5616

56

61

51

odds against rolling a 5


Odds in Favor

Odds in favor of event P event occurs

P event fails to occur

P success P failure


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


Five red apples and six yellow apples are placed in a bag. If one apple is selected at random, determine the odds in favor of the apple being red.

a. 5:11

b. 6:11

c. 5:6

d. 6:5


Probability from Odds

Example: The odds against spinning a blue on a certain spinner are 4:3. Find the probability that

a) a blue is spun. b) a blue is not spun.


Solution

Since the odds are 4:3 the denominators must be 4 + 3 = 7.

The probabilities ratios are:

( )=3blue

7P

( )=4not blue

7P

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

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