Independent and

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Independent and Dependent Events
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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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
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Objectives
Find the probability of independent events. Find
the probability of dependent events.
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Vocabulary
independent events dependent events
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Adams 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.
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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.
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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.
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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.
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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.
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To determine the probability of two independent
events, multiply the probabilities of the two
events.
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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.
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Example 2A Continued
P(red, green) P(red) ? P(green)
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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)
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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
.
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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
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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
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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.
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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?
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Example 3 Continued
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
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Example 3 Continued
Draw a diagram.
After Grant selects a bag, the sample space
changes. So the events are dependent.
After Grant selects a bag, the sample space
changes. So the events are dependent.
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Example 3 Continued
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.
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Example 3 Continued
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.
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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?
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Check It Out! Example 3 Continued
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.

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Check It Out! Example 3 Continued
Draw a diagram.
After the first selection, the sample space
changes. So the events are dependent.
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Check It Out! Example 3 Continued
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.
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Check It Out! Example 3 Continued
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.
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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.
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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?
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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?