Transcript Probability of Dependent Events

Probability
Grade 7 Pre-Algebra
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Drawing a Tree Diagram
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You can use a tree diagram to display and count
possible choices.
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Example: A school team sells caps in two colors (blue or
white), two sizes (child or adult), and two fabrics (cotton
or polyester)
cotton
child
blue
polyester
cotton
Each branch of the “tree”
represents one choice – for
example, blue-child-cotton
adult
polyester
cotton
child
white
polyester
There are 8 possible cap choices
cotton
adult
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polyester
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Your Turn!

Suppose the caps in the previous example also
came in black. Draw a tree diagram. How
many cap choices are there?
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Counting Principle
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Another way to count choices is to use the Counting
Principle.
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If there are m ways of making one choice, and n ways of
making a second choice, then there are m . n ways of the
first choice followed by the second.
The Counting principle is particularly useful when a
tree diagram would be too large to draw.
The Counting Principle is sometimes called the
“Multiplication Counting Principle”.
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Using the Counting Principle
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Example: How many two-letter monograms
are possible?
first letter
second letter
monograms
Possible choices
possible choices
possible choices
26
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x
26
=
676
There are 676 possible two-letter monograms
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Your Turn!

How many three-letter monograms are
possible?
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Theoretical Probability
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You can count outcomes to help you find the
theoretical probability of an event in which
outcomes are equally likely.
P(event) = number of favorable outcomes
number of possible outcomes
A sample space is a list of all possible
outcomes.
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Using a Tree Diagram

Example: Use a tree diagram to find the
sample space for tossing two coins. Then find
the probability of tossing two tails.
heads
heads
tails
The tree diagram shows there
are four possible outcomes, one
of which is tossing two tails.
heads
tails
tails
P(event) = number of favorable outcomes
number of possible outcomes
P(two tails) = number of two-tail outcomes
number of possible outcomes
=¼
The probability of tossing two tails is ¼ .
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Your Turn!

You toss two coins. Find P(one head and one
tail).
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Using the Counting Principle
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Example: In lotteries the winning number is made up of four digits chosen
at random. Suppose a player buys two tickets with different numbers.
What is the probability that the player has a winning ticket?
First find the number of possible outcomes. For each digit there are 10
possible outcomes, 0 through 9.
1st digit
2nd digit
3rd digit
4th digit
total
Outcomes
Outcomes
Outcomes
Outcomes
Outcomes
10
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x
10
x
10
x 10
=
10,000
Then find the probability when there are two favorable outcomes.
P(winning ticket) = number of favorable outcomes =
2
number of possible outcomes 10,000
The probability is 2/10,000 or 1/5,000
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Your Turn!

A lottery uses five digits chosen at random.
Find the probability of buying a winning
ticket.
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Independent Events
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Independent Events are events for which the occurrence of
one event does not effect the probability of the occurrence of
the other.
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Suppose there are 10 cards with one number from 1 to 10 on them.
You are interested to draw an even number and then again a second
card with even number.
If you replace your first card, the probability of getting an even
number on the second card is unaffected.
Probability of Independent Events
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For two independent events A and B, the probability of both events
occurring is the product of the probabilities of each event occurring.
P(A, then B) = P(A) x P(B)
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Finding Probability of Independent
Events
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Example: You roll a number cube once. Then you
roll it again. What is the probability that you get 2 on
the first roll and a number greater than 4 on the
second roll?
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There is one 2 among 6 numbers on a number cube.
P(2) = 1/6
P(greater than 4) = 2/6 There are two numbers greater than 4 on a number cube.
P(2, then greater than 4) = P(2) x P(greater than 4)
= 1/6 x 2/6
= 2/36, or 1/18
The probability is 1/18.
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Your Turn!
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You toss a coin twice. Find the probability of
getting two heads.
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Example
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Under the best conditions, a wild bluebonnet seed
has 20% probability of growing. If you select two
seeds at random, what is the probability that both
will grow, under best conditions?
P(a seed grows) = 20% or 0.20
P(two seeds grow) = P(a seed grows) x P(a seed grows)
= 0.20 x 0.20
= 0.04
= 4%
The probability that two seeds grow is 4%
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Your Turn!

Chemically treated bluebonnet seeds have a
30% probability of growing. You select two
such seeds at random. What is the probability
that both will grow?
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Dependent Events
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Dependent Events are events for which the occurrence of
one event affects the probability of the occurrence of the
other.
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Suppose you want to draw two even-numbered cards from cards
having numbers from 1 to 10. You draw one card. Then , without
replacing the first card, you draw a second card. The probability of
drawing an even number on the second card is affected.
Probability of Dependent Events
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For two dependent events A and B, the probability of both events
occurring is the product of the probability of the first event and the
probability that, after the first event, the second event occurs.
P(A, then B) = P(A) x P(B after A)
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Finding Probability for Dependent
Events
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Example: Three girls and two boys volunteer to represent
their class at a school assembly. The teacher selects one name
and then another from a bag containing the five students’
names. What is the probability that both representatives will
be girls?
Three of five students are girls.
P(girl) = 3/5
P(girl after girl) = 2/4
If a girl’s name is drawn, two of the four remaining
students are girls.
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P(girl, then girl) = P(girl) x P(girl after girl)
= 3/5 x 2/4
= 6/20 or 3/10
The probability that both representatives will be girls is 3/10.
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Your Turn!

In the previous example, find
P(boy, then girl)
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P(girl, then boy)
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Break!!!
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Assessment
1.
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2.
You can choose a burrito having one filling
wrapped in one tortilla. Draw a tree diagram
to count the number of burrito choices.
Tortillas: flour or corn; fillings: beef,
chicken, bean, cheese, or vegetable
There are four roads from Marsh to Taft and
seven roads from Taft to Polk. Use the
Counting Principle to find the number of
routes from Marsh to Polk through Taft.
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Assessment
3.
4.
5.
Use a tree diagram to find the sample space for
tossing three coins. Then find the probability:
P(three heads).
Use the Counting Principle to find the probability
of choosing the three winning lottery numbers when
the numbers are chosen at random from 1 to 50.
Numbers can repeat.
Find the probability. You roll two odd numbers and
pick a vowel (when you roll two number cubes and
pick a letter of the alphabet at random).
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Assessment
6.
7.
8.
You roll a number cube twice. What is the
probability that you roll 6, then 2 or 5.
Weather forecasters are accurate 91% of the time
when predicting precipitation for the day. What is
the probability that a forecaster will make correct
precipitation predictions two days in a row?
You select the card, G. Then without replacing the
card, you select R or A. Find the probability.
P
R E
A L
G E
B R
A
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Assessment
9.
10.
You roll a number cube. You roll it again. Are
the events independent or dependent?
Explain.
You pick a marble from a bag containing 1
green marble, 4 red marbles, 2 yellow
marbles, and 3 black marbles. You replace
the first marble and then select a second one.
Find the probability P(red, then yellow).
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Good Job!
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Remember to do the practice worksheets!!!
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