Section 12.6 OR and AND Problems

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

Section 12.6

OR and AND Problems


What You Will Learn

Compound ProbabilityOR ProblemsAND ProblemsIndependent Events

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Compound Probability

In this section, we learn how to solve compound probability problems that contain the words and or or without constructing a sample space.

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OR Probability

The or probability problem requires obtaining a “successful” outcome for at least one of the given events.

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Probability of A or B

To determine the probability of A or B, use the following formula.

P(A or B) P(A) P(B) P(A and B)

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Example 1: Using the Addition FormulaEach 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 hat, and one piece is randomly selected. Determine the probability that the piece of paper selected contains an even number or a number greater than 6.

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Example 1: Using the Addition FormulaSolutionDraw a VennDiagram

P(even)

5

10

P( 6)

4

10

P(both)

2

1012.6-7


Example 1: Using the Addition FormulaSolution

P

even or

> 6

P even P 6 P both

5

10

4

10

2

10

7

10The seven numbers that are even or greater than 6 are 2, 4, 6, 7, 8, 9, and 10.

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Mutually Exclusive

Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously.If two events are mutually exclusive, then the P(A and B) = 0.The addition formula simplifies to

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

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Example 3: Probability of A or BOne card is selected from a standard deck of playing cards. Determine whether the following pairs of events are mutually exclusive and determineP (A or B).a) A = an ace, B = a 9SolutionImpossible to select both so

P ace or 9 P ace P 9

4

52

4

52

2

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Example 3: Probability of A or Bb) A = an ace, B = a heartSolutionPossible to select the ace of hearts, so NOT mutually exclusive

P

ace or

heart

P ace P heart P both

4

52

13

52

1

52

16

52

4

13

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Example 3: Probability of A or Bc) A = a red card, B = a black cardSolutionImpossible to select both so mutually exclusive

P red or black P red P black

26

52

26

52

52

52 1

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Example 3: Probability of A or Bd) A = a picture card, B = a red cardSolutionPossible to select a red picture card, so NOT mutually exclusive

P

picture card

or red card

P

picture

card

P

red

card

P both

12

52

26

52

6

52

32

52

8

13

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And Problems

The and probability problem requires obtaining a favorable outcome in each of the given events.

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Probability of A and B

To determine the probability of A and B, use the following formula.

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

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Probability of A and BSince we multiply to find P (A and B), this formula is sometimes referred to as the multiplication formula.When using the multiplication formula, we always assume that event A has occurred when calculating P(B) because we are determining the probability of obtaining a favorable outcome in both of the given events.

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Example 5: An Experiment without ReplacementTwo cards are to be selected without replacement from a deck of cards. Determine the probability that two spades will be selected.

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Example 5: An Experiment without ReplacementSolutionThe probability of selecting a spade on the first draw is 13/52.Assuming we selected a spade on the first draw, then the probability of selecting a spade on the second draw is 12/51.

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Example 5: An Experiment without ReplacementSolution

P(2 spades) P(spade 1) P(spade 2)

13

5212

51

1

4

4

17

1

17

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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 occurrence of the other event.

Rolling dice and tossing coins are examples of independent events.

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Example 6: Independent or Dependent Events?One hundred people attended a charity benefit to raise money for cancer research. Three people in attendance will be selected at random without replacement, and each will be awarded one door prize. Are the events of selecting the three people who will be awarded the door prize independent or dependent events?

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Example 6: Independent or Dependent Events?SolutionThe events are dependent since each time one person is selected, it changes the probability of the next person being selected.P(person A is selected) = 1/100If person B is actually selected, then on the second drawing,P(person A is selected) = 1/99

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Independent or Dependent Events?

In general, in any experiment in which two or more items are selected without replacement, the events will be dependent.

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Documents

Section 12.6 OR and AND Problems