Download 5.3 ADDITION RULE AND MULTIPLICATION RULE Textbook

SECTION 5.3
Addition Rule and Multiplication Rule 207
5.3 ADDITION RULE AND MULTIPLICATION RULE
Textbook Reference Sections 11.4, 11.5
CLAST OBJECTIVES
" Identify the probability of a specified outcome
" Solve real-world problems involving probability
Read “the probability of A or B is equal to
Addition Rule
the probability of A plus the probability of
P ( A or B ) = P ( A ) + P ( B ) – P ( A and B )
B minus the probability of A and B.”
Mathematical Notation:
P(A U B) = P ( A ) + P ( B ) – P ( A ∩ B )
Event A
Events
A and B
Examples
a) A die is rolled. What is the probability
of getting an even number or a number
less than 3 ?
b) There are ten applicants with equal
qualifications. Four are females. Six of the
applicants have military experience. Four
of the six are males. One of the ten
applicants is given the job. What is the
probability that the applicant is female or
has military experience?
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Event B
Solutions
There are six possible outcomes.
Let Event A be getting an even number.
A ={2,4,6}
Let Event B = the number less than 3.
B={1,2}
3
2
P(A)=
P(B)=
6
6
P( A and B ) is the probability of an even
number and that even number is less than 3.
1
P( A and B ) =
6
P( A or B ) = P ( A ) + P ( B ) – P ( A and B )
3 2 1 4 2
= + − = =
6 6 6 6 3
There are ten possible outcomes.
Let Event A be getting a female applicant.
Let Event B be getting an applicant with
military experience.
4
6
P(B)=
P(A)=
10
10
P ( A and B ) is the probability of choosing an
applicant who is a female with military
2
experience. P( A and B ) =
10
P ( A or B ) = P ( A ) + P ( B ) – P ( A and B )
4
6
2
8 4
P ( A or B ) =
+
−
=
=
10 10 10 10 5
208 CHAPTER 5
Probability and Statistics
Example
c) Consider the responses to a question
asked to 110 people.
Yes
No
Totals
Men
20
30
50
Women
10
50
60
Solution
Totals
30
80
110
Find each of the following.
i)
What is the probability of
choosing a man?
ii)
What is the probability of
choosing someone who said
“yes” ?
iii)
What is the probability of
choosing a man or someone
who said “yes” ?
i) P ( Man ) =
50
110
ii) P ( Yes ) =
30
110
iii) P ( Man or Yes )
= P ( Man ) + P ( Yes ) – P ( Man ∩ Yes )
=
50
30
20
60
6
+
−
=
=
110 110 110 110 11
Check Your Progress 5.3
1. What is the probability of getting a ‘ 1 ’ or a ‘ 4 ’ in one roll of a die?
2. An urn contains 5 red, 3 green, and 4 white balls. If one ball is drawn at random, what
is the probability of getting a red or a white ball?
3. 57 % of the children in Florida own a bicycle. 63 % own a Nintendo system. 28 % of
the children in Florida own both, a bicycle and a Nintendo system. What is the
probability that a randomly selected child will own either a bicycle or a Nintendo system?
4. Consider the following results of a survey. Find the probability of randomly selecting
a woman or a non-smoker.
Smoke
Do Not Smoke
Totals
Men
60
50
110
Women Totals
50
110
30
80
80
190
5. Using the table above, find the probability of randomly selecting a man or a smoker.
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SECTION 5.3
Addition Rule and Multiplication Rule 209
Multiplication Rule
P ( A and B ) = P ( A ) × P ( B ) if Events A and B are independent.
P ( A and B ) = P ( A ) × P( B | A ) if Events A and B are dependent.
Read “the probability of Event B occurring
given that Event A has already occurred.
Note
•
•
With Replacement: All information given remains the same for the entire calculation.
With Replacement Scenario: Ten marbles are in an urn. Two marbles will be drawn,
one at a time. After the first marble is drawn, it is put back into the urn before the
second marble is drawn. Events are independent.
Without Replacement: Information given may change during the calculation.
Without Replacement Scenario: Ten marbles are in an urn. Two marbles will be
drawn, one at a time. After the first marble is drawn, it is not put back into the urn
before the second marble is drawn. Events are dependent.
Examples
d) Consider the responses to a question
asked to 110 people.
Yes
No
Totals
Men
20
30
50
Women
10
50
60
Totals
30
80
110
Find each of the following.
i)
What is the probability of
choosing a man?
ii)
What is the probability of
choosing a man given that the
response was “yes”?
iii)
Given that a man was selected,
find the probability that he said
“yes”.
e) An urn contains 2 red balls, 4 green, 4
yellow, 3 orange, and 2 purple balls. Two
balls are drawn, one at a time without
replacement. Find the probability of
drawing a green ball and an orange ball.
Solutions
50
i) P ( Man ) =
110
ii) P ( Man | Yes )
Note that 30 people responded “yes”. Of the
30 who said “yes”, 20 were men. Hence,
20 2
=
P ( Man | Yes ) =
30 3
iii) P ( Yes | Man )
Note that 50 were men. Of those 50 men, 20
of them said “yes”. Hence,
20 2
=
P ( Man | Yes ) =
50 5
There are 15 possible outcomes. What you
get on the first draw reduces the possible
outcomes for the second draw by one since
the ball on the first draw is not replaced.
Event A: Getting a green ball
Event B: Getting an orange ball given the
first ball was green
P(A and B) = P(A) × P(B|A)
=
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4 3
12
2
× =
=
15 14 210 35
210 CHAPTER 5
Probability and Statistics
Examples
f) A coin is tossed three times. What is the
probability of getting three heads?
g) An urn contains 2 red balls, 4 green, 4
yellow, 3 orange, and 2 purple balls. Two
balls are drawn, one at a time with
replacement. Find the probability of
drawing a green ball and an orange ball.
Solutions
There are only two possible outcomes in a
coin toss, { heads , tails }. Results are
independent. What you get on first toss has
nothing to do with what’s available for the
second toss and so on.
Event A: Getting a head
Event B: Getting a head
Event C: Getting a head
P ( A and B and C ) = P(A) × P(B) × P(C)
1 1 1 1
= × × =
2 2 2 8
Again, there are 15 possible outcomes.
What you get on the first draw has no
bearing on the outcomes for the second draw
since the ball on the first draw is replaced.
Event A: Getting a green ball
Event B: Getting an orange ball
P(A and B) = P(A) × P(B)
4 3
12
4
=
× =
=
15 15 225 75
Check Your Progress 5.3
6. A coin is tossed twice. What is the probability of getting a head first, and then a tail?
7. A coin is tossed four times. What is the probability of getting four heads in succession?
8. A fair die is rolled three times. What is the probability of getting a ‘ 6 ’, then a ‘ 3 ’,
and then a ‘ 1 ’ ?
9. Consider the following results of a survey. Given that a non-smoker was selected, what
is the probability that the person is a woman?
Men
Women Totals
Smoke
60
50
110
Do Not Smoke
50
30
80
Totals
110
80
190
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SECTION 5.3
Addition Rule and Multiplication Rule 211
10. A box contains ten batteries of which three are defective. If two batteries are drawn
in succession without replacement, what is the probability of getting two non-defective
batteries?
11. A gym teacher has a bag that contains 2 basketballs, 1 football, 2 soccerballs, and 3
volleyballs. Three balls are tossed out, one at a time. Find the probability that the first
ball is a soccerball, the second one is a basketball, and the third one is a volleyball.
12. An urn contains 2 white balls, 3 black, 4 blue, 3 green, and 2 pink balls. Two balls
are drawn, one at a time with replacement. Find the probability of drawing a blue ball and
white ball.
For Questions 13 – 16, consider the results of a survey of 150 apartments.
1 bedroom
2 bedrooms
3 bedrooms
st
20
40
10
1 Floor
30
35
15
2 nd Floor
13. What is the probability of randomly selecting a two-bedroom apartment?
14. What is the probability of selecting a second floor apartment given that it has three
bedrooms?
15. What is the probability of selecting a one-bedroom apartment or an apartment on the
first floor?
16. What is the probability of selecting an apartment with at least two bedrooms?
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212 CHAPTER 5
Probability and Statistics
See If You
Remember
SECTIONS 5.1 & 5.2
1. Jane has 5 blouses, 3 skirts, and 2 belts. How many outfits can Jane make?
2. How many ways can we make up a 5 - digit code if digits can be repeated?
3. Evaluate 9 P 2 .
4. Evaluate 9 C 2 .
5. A bag has 10 different candies. Mary wants 4 candies. How many selections can she
make?
6. The probability of picking a black ball is 0.79. What is the probability of not picking a
black ball?
7. A die is rolled. What are the odds against getting a number greater than 5?
8. The odds in favor of a team winning are 7 to 5. What is the probability that the team
will lose?
9. How many ways can six books be placed on a shelf?
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