Section 12.2 Theoretical Probability

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 12.2

Theoretical Probability

12.2-1


What You Will Learn

Equally Likely OutcomesTheoretical Probability

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Equally Likely OutcomesIf 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 outcomes12.2-3


Example 1: Determining Probabilities

A die is rolled. Find the probability of rollinga) a 5.b) an even number.c) a number greater than 3.d) a 7.e) a number less than 7.

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Example 1: Determining ProbabilitiesSolutiona)

b) Rolling an even number can occur in three ways: 2, 4 or 6.

P(5)

number of outcomes that will result in a 5

total number of possible outcomes

1

6

P(rolling an even number)

3

6

1

2

12.2-5


Example 1: Determining ProbabilitiesSolution

c) Three numbers are greater than 3: 4, 5 or 6.

P(greater than 3)

3

6

1

2

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Example 1: Determining ProbabilitiesSolution

d) No outcomes will result in a 7. Thus, the event cannot occur and the probability is 0.

P(7)

0

60

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Solutione) All the outcomes 1 through 6 are less than 7. Thus, the event must occur and the probability is 1.

P(number less than 7)

6

61

Example 1: Determining Probabilities

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Important Probability FactsThe 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.

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The Sum of the Probabilities Equals 1

P(A) + P(not A) = 1

or

P(not A) = 1 – P(A)

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Example 3: Selecting One Card from a DeckA standard deck of 52 playing cards is shown.

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Example 3: Selecting One Card from a DeckThe deck consists of four suits: hearts, clubs, diamonds, and spades. Each suit has 13 cards, including numbered cards ace (1) through 10 and three picture (or face) cards, the jack, the queen, and the king.

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Example 3: Selecting One Card from a DeckHearts and diamonds are red cards; clubs and spades are black cards. There are 12 picture cards, consisting of 4 jacks, 4 queens, and 4 kings. One card is to be selected at random from the deck of cards. Determine the probability that the card selected is

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Example 3: Selecting One Card from a Decka)a 7.b)not a 7.c)a diamond.d)a jack or queen or king (a

picture card).e)a heart and spade.f)a card greater than 6 and less

than 9.12.2-14


Example 3: Selecting One Card from a DeckSolutiona)a 7. There are 4 7’s in a deck

of cards.

b)not a 7.

P(7)

4

52

1

13

P(not a 7) 1 P(7) 1

1

13

12

1312.2-15


Example 3: Selecting One Card from a DeckSolutionc)a diamond.

There are 13 diamonds in the deck.

P(7)

13

52

1

4

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Example 3: Selecting One Card from a DeckSolutiond)a jack or queen or king (a

picture card).There are 4 jacks, 4 queens, and 4 kings or a total of 12 picture cards.

P(7)

12

52

3

13

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Example 3: Selecting One Card from a DeckSolutione)a heart and spade.

The word and means both events must occur. This is not possible, that one card is both, the probability = 0.

P(heart and spade)

0

520

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Example 3: Selecting One Card from a DeckSolutionf)a card greater than 6 and less

than 9.The cards that are both greater than 6 and less than 9 are 7’s and 8’s. There are 4 7’s and 4 8’s, or 8 total.

P( 6 and < 9)

8

52

2

13

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Section 12.2 Theoretical Probability