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Independent
Independentand
andDependent
10-7Events
10-7
Dependent Events
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 1Algebra
Algebra11
Holt
McDougal
Independent and Dependent
10-7 Events
Warm Up
Find the theoretical probability of each
outcome
1. rolling a 6 on a number cube.
2. rolling an odd number on a number cube.
3. flipping two coins and both landing head
up
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Objectives
Find the probability of independent
events.
Find the probability of dependent
events.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Vocabulary
independent events
dependent events
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Adam’s teacher gives the class two list of titles and
asks each student to choose two of them to read.
Adam can choose one title from each list or two
titles from the same list.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Events are independent events if the occurrence
of one event does not affect the probability of the
other. Events are dependent events if the
occurrence of one event does affect the probability
of the other.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 1: Classifying Events as Independent or
Dependent
Tell whether each set of events is independent
or dependent. Explain you answer.
A. You select a card from a standard deck of
cards and hold it. A friend selects another
card from the same deck.
Dependent; your friend cannot pick the card you
picked and has fewer cards to choose from.
B. You flip a coin and it lands heads up. You flip
the same coin and it lands heads up again.
Independent; the result of the first toss does not
affect the sample space for the second toss.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Check It Out! Example 1
Tell whether each set of events is independent
or dependent. Explain you answer.
a. A number cube lands showing an odd
number. It is rolled a second time and
lands showing a 6.
Independent; the result of rolling the number
cube the 1st time does not affect the result of the
2nd roll.
b. One student in your class is chosen for a
project. Then another student in the class
is chosen.
Dependent; choosing the 1st student leaves fewer
students to choose from the 2nd time.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Suppose an experiment involves flipping two fair
coins. The sample space of outcomes is shown by
the tree diagram. Determine the theoretical
probability of both coins landing heads up.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Now look back at the separate theoretical
probabilities of each coin landing heads up.
The theoretical probability in each case is .
The product of these two probabilities is
, the same probability shown by the tree
diagram.
To determine the probability of two independent
events, multiply the probabilities of the two
events.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 2A: Finding the Probability of Independent
Events
An experiment consists of randomly selecting a
marble from a bag, replacing it, and then
selecting another marble. The bag contains 3
red marbles and 12 green marbles. What is the
probability of selecting a red marble and then a
green marble?
Because the first marble is replaced after it is
selected, the sample space for each selection is the
same. The events are independent.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 2A Continued
P(red, green) = P(red)  P(green)
The probability of selecting red
is
, and the probability of
selecting green is
Holt McDougal Algebra 1
.
Independent and Dependent
10-7 Events
Example 2B: Finding the Probability of Independent
Events
A coin is flipped 4 times. What is the
probability of flipping 4 heads in a row.
Because each flip of the coin has an equal
probability of landing heads up, or a tails, the
sample space for each flip is the same. The events
are independent.
P(h, h, h, h) = P(h) • P(h) • P(h) • P(h)
The probability of landing
heads up is with
each event.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Check It Out! Example 2
An experiment consists of spinning the
spinner twice. What is the probability of
spinning two odd numbers?
The result of one spin does
not affect any following
spins. The events are
independent.
With 6 numbers on the spinner, 3 of which are
odd, the probability of landing on two odd
numbers is
.
P(odd, odd) = P(odd) • P(odd)
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Suppose an experiment involves drawing marbles
from a bag. Determine the theoretical probability of
drawing a red marble and then drawing a second
red marble without replacing the first one.
Probability of drawing a red marble on the first draw
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Suppose an experiment involves drawing marbles
from a bag. Determine the theoretical probability of
drawing a red marble and then drawing a second
red marble without replacing the first one.
Probability of drawing a red marble on the second
draw
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
To determine the probability of two dependent
events, multiply the probability of the first event
times the probability of the second event after the
first event has occurred.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 3: Application
A snack cart has 6 bags of pretzels and 10
bags of chips. Grant selects a bag at
random, and then Iris selects a bag at
random. What is the probability that
Grant will select a bag of pretzels and Iris
will select a bag of chips?
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 3 Continued
1
Understand the Problem
The answer will be the probability that a bag of
chips will be chosen after a bag of pretzels is
chosen.
List the important information:
• Grant chooses a bag of pretzels from 6 bags
of pretzels and 10 bags of chips.
• Iris chooses a bag of chips from 5 bags of
pretzels and 10 bags of chips.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 3 Continued
2
Make a Plan
Draw a diagram.
After Grant selects a bag, the sample space
changes. So the events are dependent.
Iris chooses from:
Grant chooses from:
pretzels
chips
After Grant selects a bag, the sample space
changes. So the events are dependent.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 3 Continued
3
Solve
P(pretzel and chip) = P(pretzel) • P(chip after pretzel)
Grant selects one of 6 bags of
pretzels from 16 total bags.
Then Iris selects one of 10
bags of chips from 15 total
bags.
The probability that Grant selects a bag of
pretzels and Iris selects a bag of chips is .
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Example 3 Continued
4
Look Back
Drawing a diagram helps you see how the
sample space changes. This means the
events are dependent, so you can use the
formula for probability of dependent events.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Check It Out! Example 3
A bag has 10 red marbles, 12 white
marbles, and 8 blue marbles. Two
marbles are randomly drawn from the
bag. What is the probability of drawing
a blue marble and then a red marble?
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Check It Out! Example 3 Continued
1
Understand the Problem
The answer will be the probability that a
red marble will be chosen after a blue
marble is chosen.
List the important information:
• A blue marble is chosen from a bag containing
10 red, 12 white, and 8 blue marbles.
• Then a red marble is chosen from a bag of
10 red, 12 white, and 7 blue marbles.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Check It Out! Example 3 Continued
2
Make a Plan
Draw a diagram.
First choice from:
Second choice from:
After the first selection, the sample space
changes. So the events are dependent.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Check It Out! Example 3 Continued
3
Solve
P(blue and red) = P(blue) • P(red after blue)
One of 8 blue marbles is
selected from a total of 30
marbles. Then one of 10 red
marbles is selected from the
29 remaining marbles.
The probability that first a blue marble is
selected and then a red marble is selected is
Holt McDougal Algebra 1
.
Independent and Dependent
10-7 Events
Check It Out! Example 3 Continued
4
Look Back
Drawing a diagram helps you see how the
sample space changes. This means the
events are dependent, so you can use the
formula for probability of dependent events.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Lesson Quiz: Part I
Tell whether each set of events is independent or
dependent. Explain your answer.
1. flipping two different coins and each coin
landing showing heads
Independent; the flip of the first coin does not
affect the sample space for the flip of the second
coin.
2. drawing a red card from a standard deck of cards
and not replacing it; then drawing a black card
from the same deck of cards
Dependent; there are fewer cards to choose
from when drawing the black card.
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Lesson Quiz: Part II
3. Eight cards are numbered from 1 to 8 and placed
in a box. One card is selected at random and not
replaced. Another card is randomly selected.
What is the probability that both cards are
greater than 5?
4. An experiment consists of randomly selecting a
marble from a bag, replacing it, and then
selecting another marble. The bag contains 3
yellow marbles and 2 white marbles. What is the
probability of selecting a white marble and then
a yellow marble?
Holt McDougal Algebra 1
Independent and Dependent
10-7 Events
Lesson Quiz: Part III
5. A number cube is rolled two times. What is
the probability of rolling an even number first
and then a number less than 3?
Holt McDougal Algebra 1