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Section 12.6 OR and AND Problems. What You Will Learn. Compound Probability OR Problems AND Problems Independent Events. 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. - PowerPoint PPT Presentation
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 12.6
OR and AND Problems
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Compound ProbabilityOR ProblemsAND ProblemsIndependent Events
12.6-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
OR Probability
The or probability problem requires obtaining a “successful” outcome for at least one of the given events.
12.6-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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)
12.6-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Using the Addition FormulaSolutionDraw a VennDiagram
P(even)
5
10
P( 6)
4
10
P(both)
2
1012.6-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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).
12.6-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
1312.6-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.6-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.6-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.6-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
And Problems
The and probability problem requires obtaining a favorable outcome in each of the given events.
12.6-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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)
12.6-15
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: An Experiment without ReplacementSolution
P(2 spades) P(spade 1) P(spade 2)
13
5212
51
1
4
4
17
1
17
12.6-19
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
12.6-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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?
12.6-21
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
12.6-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Independent or Dependent Events?
In general, in any experiment in which two or more items are selected without replacement, the events will be dependent.
12.6-23