Download (A or B).

Section 12.6
OR and AND
Problems
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What You Will Learn
Compound Probability
OR Problems
AND Problems
Independent Events
12.6-2
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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.
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
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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)
12.6-5
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Example 1: Using the Addition
Formula
Each 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
Formula
Solution
Draw a Venn
Diagram
5
P(even) 
10
4
P( 6) 
10
2
P(both) 
10
12.6-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Using the Addition
Formula
Solution
 even or 
P
 P even  P  6  P both

> 6

5
4
2
7




10 10 10
10
The seven numbers that are even or
greater than 6 are 2, 4, 6, 7, 8, 9, and
10.

12.6-8
   
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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).
12.6-9
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Example 3: Probability of A or B
One card is selected from a standard
deck of playing cards. Determine
whether the following pairs of events
are mutually exclusive and determine
P (A or B).
a) A = an ace, B = a 9
Solution
Impossible to select both so
4
4
2
P ace or 9  P ace  P 9 


52 52 13

12.610
   
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Example 3: Probability of A or B
b) A = an ace, B = a heart
Solution
Possible to select the ace of hearts, so
NOT mutually exclusive
 ace or 
P
 P ace  P heart  P both

 heart 
4 13 1
16
4





52 52 52 52 13
  
12.611
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 

Example 3: Probability of A or B
c) A = a red card, B = a black card
Solution
Impossible to select both so mutually
exclusive

   
P red or black  P red  P black
26 26
52



52 52
52
12.612
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
1
Example 3: Probability of A or B
d) A = a picture card, B = a red card
Solution
Possible to select a red picture card, so
NOT mutually exclusive
 picture card
 picture
 red 
P

P

P

P
both
 card 
 card
 or red card 

12 26 6
32
8





52 52 52 52 13
12.613
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
And Problems
The and probability problem
requires obtaining a favorable outcome
in each of the given events.
12.614
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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)
12.615
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Probability of A and B
Since 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.
12.616
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Example 5: An Experiment
without Replacement
Two cards are to be selected without
replacement from a deck of cards.
Determine the probability that two
spades will be selected.
12.617
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Example 5: An Experiment
without Replacement
Solution
The 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.
12.618
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Example 5: An Experiment
without Replacement
Solution
P(2 spades)  P(spade 1)  P(spade 2)
13 12


52 51
1 4
 
4 17
12.619
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.
12.620
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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?
12.621
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Example 6: Independent or
Dependent Events?
Solution
The 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) = 1/100
If person B is actually selected, then
on the second drawing,
P(person A is selected) = 1/99
12.622
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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.
12.623
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