Page 1 of 6
Independent and
Dependent Events
BEFORE
Now
WHY?
You found the probability of
disjoint events.
You’ll find the probability of
dependent events.
So you can find the probability of
choosing socks, as in Example 3.
You can find the probability of two events occurring under
different circumstances.
Word Watch
1 A bag contains 9 pieces of paper, with 5 pieces having
an O and 4 pieces having an X. Suppose you randomly
choose a piece of paper from the bag, you get an O,
and you don’t put it back. Then you randomly choose a
second piece of paper. What is the probability that the
second piece of paper has an X?
independent events, p. 664
dependent events, p. 664
2 Suppose that you repeat Step 1, but this time the first piece of paper has an X.
What is the probability that the second piece of paper also has an X?
3 Why do you think the probabilities in Steps 1 and 2 are different?
Two events are independent events if the occurrence of one event
does not affect the likelihood that the other event will occur. Two events
are dependent events if the occurrence of one event does affect the
likelihood that the other will occur.
EXAMPLE
with
1
Independent and Dependent Events
In Step 1 of the activity, you chose an O first, then an X. Are these
events independent or dependent?
Solving
When you consider the
outcomes of two events,
the events are called
compound events.
Independent, dependent,
and disjoint events are
compound events.
As Step 2 of the activity showed, whether or not you choose an O first
does affect the likelihood that you choose an X second. This is because
the ratio of X’s to O’s in the bag changes after the first piece of paper is
chosen and not put back.
ANSWER The events are dependent.
Your turn now
A jar contains 5 red and 7 blue marbles.
1. You randomly choose a marble, put it back, then randomly choose
another marble. Are the events “choose a red marble first” and
“choose a blue marble second” independent or dependent?
664
Chapter 13
Probability
Page 2 of 6
In common usage, being
independent means being
free from the control of
others. This may help you
remember the meaning of
independent events.
1p3
2p6
P(H and odd) 1 3
p 2 6
P(H) p P(odd)
T
1
H, 1
T, 1
2
H, 2
T, 2
3
H, 3
T, 3
4
H, 4
T, 4
5
H, 5
T, 5
6
H, 6
T, 6
This result suggests the following rule.
Probability of Independent Events
Words For two independent events, the probability that both
events occur is the product of the probabilities of the events.
Algebra If A and B are independent events, then
P(A and B) P(A) p P(B).
EXAMPLE
2
Probability of Independent Events
BAN
KRU
PT
1 Find the probability of each event.
2
8
P($200) 0.25
$
$
3
2
0
0
0 0
$
1
0
0
Game Show As a contestant on a game show,
you need to spin the money wheel at the right,
which is divided into equal sections. Find the
probability that you get $200 on your first spin
and go bankrupt on your second spin.
$
1
0
0
Vocabulary
H
LOS
E TU
RN
$
$
3
2
0
0
0 0
with
Independent Events A coin is flipped and a number
cube is rolled. The table of outcomes helps you see the
relationship between the probability of two events
together and the probabilities of the individual events.
“$200” appears 2 times.
1
8
P(bankrupt) 0.125
“Bankrupt” appears once.
2 Because the events are independent, multiply the probabilities.
P($200 and bankrupt) P($200) P(bankrupt)
0.25 0.125
0.03125
ANSWER The probability that you get $200 on your first spin and go
bankrupt on your second spin is 0.03125, or about 3%.
Lesson 13.6
Independent and Dependent Events
665
Page 3 of 6
Dependent Events If A and B are dependent events, the probability that
B occurs given A is not the same as the probability of B. So, you should
use P(B given A) instead of P(B) to represent the probability that B will
occur given that A has occurred.
Probability of Dependent Events
Words For two dependent events, the probability that both
events occur is the product of the probability of the first
event and the probability of the second event given the first.
Algebra If A and B are dependent events, then
P(A and B) P(A) p P(B given A).
EXAMPLE
3
Probability of Dependent Events
Socks A drawer has 12 white, 7 black, and 6 striped socks. You randomly
choose 1 sock from the drawer, then randomly choose another sock
without replacing the first. Find the probability that both are white.
Solution
Find the probability of the first event and the probability of the second
event given the first. Then multiply the probabilities.
12
25
1 P(white) Of the 25 socks, 12 are white.
11
Of the remaining 24 socks,
11 are white.
2 P(white given white) 24
12
11
3 P(white and white) 25
24
Multiply probabilities.
1
12 11
25 24
Divide out common factor.
11
50
Multiply.
2
11
50
ANSWER The probability that both socks are white is , or 22%.
Your turn now
Refer to Example 3.
2. Find the probability that both socks are black when the first sock
chosen is not replaced.
666
Chapter 13
Probability
Page 4 of 6
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More Practice, p. 717
Getting Ready to Practice
1. Vocabulary Explain the difference between independent and
dependent events.
Events A and B are independent events. Find P(A and B).
2. P(A) 0.3
P(B) 0.7
3. P(A) 0.5
4. P(A) 0.8
P(B) 0.5
P(B) 0.2
Events A and B are dependent events. Find P(A and B).
5. P(A) 0.9
P(B given A) 0.8
6. P(A) 0.6
7. P(A) 0.25
P(B given A) 0.25
P(B given A) 0.2
8. Baseball You are watching a baseball game. Tell whether the events
are independent or dependent.
Event A: The third batter in the lineup hits a home run.
Event B: The fourth batter in the lineup hits a home run.
Practice and Problem Solving
with
Example
1
2
3
Homework
Exercises
9–10
11–13, 18–20
14–16, 21–22
Online Resources
Tell whether the events are independent or dependent.
9. You roll a number cube and get a 5, and you flip a coin and get tails.
10. Your CD player has a random play button that chooses songs at
random and plays each song exactly once before repeating. While
listening to a CD in random play, you hear track 3 first and track 1
second.
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• More Examples
• eTutorial Plus
Events A and B are independent events. Find the unknown
probability.
11. P(A) 0.6
P(B) 0.3
P(A and B) _?_
12. P(A) 0.2
P(B) _?_
P(A and B) 0.08
13. P(A) _?_
P(B) 0.5
P(A and B) 0.35
Events A and B are dependent events. Find the unknown probability.
14. P(A) 0.25
P(B given A) 0.5
P(A and B) _?_
15. P(A) 0.4
P(B given A) _?_
P(A and B) 0.36
16. P(A) _?_
P(B given A) 0.3
P(A and B) 0.09
17. Writing Describe a way to randomly choose one
of the 6 lettered tiles, then another, so that the
events are independent.
Lesson 13.6
Independent and Dependent Events
667
Page 5 of 6
Shoes The tables give data about the shoes
manufactured at a factory during a day.
Assuming that the events are independent, find
the probability that a randomly chosen pair of
shoes has the given description.
18. Men’s athletic shoes
19. Women’s casual shoes
20. Men’s casual shoes
Gender
Men’s
Women’s
Percent
46%
54%
Shoe Style
Athletic
Casual
Dress
Percent
22%
61%
17%
Each whole number from 1 through 10 is written on a separate
piece of paper. You randomly choose numbers one at a time, but
you do not replace them. Find the probability that both events A
and B will occur.
21. Event A: The first number you choose is an odd number.
Event B: The second number you choose is an odd number.
22. Event A: The first number you choose is a 2.
Event B: The second number you choose is an even number.
In Exercises 23 and 24, tell whether the situation describes
independent events or dependent events. Then answer the
question.
Art
23. Gumballs A bag contains 24 blue, 20 green, and 16 yellow gumballs.
You randomly choose a gumball from the bag, and you do not replace
it. Then you randomly choose another gumball. What is the probability
that both gumballs are green?
24. Shopping The table shows the T-shirts on a
■
clearance rack at a clothing store. Suppose that
you randomly choose one T-shirt and put it back.
Then you randomly choose a second T-shirt.
What is the probability that the first T-shirt is a
small and the second T-shirt is a large?
Gumballs
The elephant shown above is
part of a gumball mural by
Franz Spohn. He used
14,300 gumballs to create
the rectangular mural, which
is 8 feet long and 6 feet
wide. How many gumballs are
in 1 square foot of the
mural? Round your answer to
the nearest whole number.
668
Chapter 13
Probability
Size
Number
Small
12
Medium
24
Large
14
25. Perform an Experiment First, roll 2 number cubes 25 times. For each
roll, record whether you get 2 of a kind. What is the experimental
probability of getting 2 of a kind? Now, suppose you roll the 2 number
cubes 2 more times. Use your experimental probability from the first
25 rolls to find the probability of getting 2 of a kind for both of the rolls.
26. Challenge The table shows the size
and color of paper clips in a box. You
randomly choose paper clips one at a
time from the box, but you do not replace
them. What is the probability that the first
three paper clips that you choose are
small and yellow?
Small
Large
Red
10
10
Blue
10
10
Yellow
15
15
Page 6 of 6
Mixed Review
Write the decimal as a percent. (Lesson 9.3)
27. 0.364
28. 0.0048
29. 2.35
In Exercises 31 and 32, use the table at the right,
which shows the results of a survey that asked
students their favorite type of pet. (Lesson 13.5)
31. What is the probability that a randomly selected
student voted for a cat or a bird?
32. What is the probability that a randomly selected
30. 0.10006
Pet
Percent
Dog
36%
Cat
31%
Bird
12%
Other
21%
student voted for a dog or a cat?
Basic Skills Find the product.
33. 5 1.7
34. 6.8 4.1
35. 8.44 2.5
36. 9.9 3.33
Test-Taking Practice
37. Extended Response Each of the letters in the word PROBABILITY
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is written on a separate piece of paper and put in a bag. Which of
the methods of selecting two letters will have a greater probability
of getting two B’s? Explain your reasoning.
Method 1: You randomly choose a letter from the bag, but you don’t
replace it. Then you randomly choose another letter.
Method 2: You randomly choose a letter from the bag and replace it.
Then you randomly choose another letter.
What’s in the bag?
A bag contains blue, red, orange, and green cubes, and there are
50 cubes in the bag. The probability of randomly choosing each
cube is listed below. How many cubes of each color are in the bag?
14%
42%
26%
18%
Lesson 13.6
Independent and Dependent Events
669