Transcript Theoretical probability

10-4 Theoretical Probability
Warm Up
1. What is the probability of getting two
tails if two coins are tossed? 1
4
2. Give the probability that the roll of a
number cube will show 1 or 4. 1
3
3. Give the expected number of rolls that
will result in a 2 if a number cube is rolled
42 times. 7
Course 2
10-4 Theoretical Probability
Learn to find the theoretical probability of
an event.
Course 2
10-4 Theoretical
Insert Lesson
Title Here
Probability
Vocabulary
favorable outcome
theoretical probability
fair
Course 2
Theoretical
Probability
10-4 Theoretical
Probability
In the game of Scrabble®, players use tiles
bearing the letters of the alphabet to form
words. Of the 100 tiles used in a Scrabble
game, 12 have the letter E on them. What
is the probability of drawing an E from a
bag of 100 Scrabble tiles?
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Theoretical
Probability
10-4 Theoretical
Probability
In this case, pulling an E from the bag is called a
favorable outcome. A favorable outcome is an
outcome that you are looking for when you
conduct an experiment.
To find the probability of drawing an E, you can
draw tiles from a bag and record your results, or
you can find the theoretical probability.
Theoretical probability is used to estimate the
probability of an event when all the outcomes are
equally likely.
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10-4 Theoretical Probability
THEORETICAL PROBABILITY
probability =
number of favorable outcomes
total number of possible outcomes
If each possible outcome of an experiment is equally
likely, then the experiment is said to be fair.
Experiments involving number cubes and coins are
usually assumed to be fair.
You can write probability as a fraction, a
decimal, or a percent.
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10-4 Theoretical Probability
Additional Example 1A: Finding Theoretical
Probability
Find the probability. Write your answer as a
fraction, as a decimal, and as a percent
A. Andy has 20 marbles in a bag. Of these, 9 are
clear. What is the probability of drawing a clear
marble from the bag?
number of favorable outcomes
P = total
number of possible outcomes
of clear marbles Write the ratio.
P(clear) = number
total number of marbles
9
Substitute.
= 20
= 0.45 = 45% Write as a decimal and write as a
percent.
The theoretical probability of drawing a clear marble is
9 , 0.45, or 45%.
20
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10-4 Theoretical Probability
Additional Example 1B: Finding Theoretical
Probability
B. What is the probability of rolling a number less
than 4 on a fair number cube.
For a fair number cube, each of the six possible outcomes
is equally likely. There are 3 ways to roll a number less
than 4: 1, 2, or 3.
number of favorable outcomes
P = total
number of possible outcomes
P(number less than 4) = 3 numbers less than 4
6 possible outcomes
= 3
6
= 1
2
= 0.5 = 50%
The theoretical probability of rolling a number less than 4 is
1 , 0.50, or 50%.
2
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10-4 Theoretical Probability
Try This: Example 1A
Find the probability. Write your answer as a
fraction, as a decimal, and as a percent.
A. Jane has 20 marbles in a bag. Of these 8 are green.
What is the probability of drawing a green marble
from the bag?
number of favorable outcomes
P = total
number of possible outcomes
of green marbles Write the ratio.
P(green) = number
total number of marbles
8
Substitute.
= 20
= 0.4 = 40% Write as a decimal and write as a
percent.
The theoretical probability of drawing a green marble is
8 , 0.4, or 40%.
20
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10-4 Theoretical Probability
Try This: Example 1B
B. What is the probability of rolling a number more
than 4 on a fair number cube.
For a fair number cube, each of the six possible outcomes
is equally likely. There are 2 ways to roll a number greater
than 4: 5 or 6.
number of favorable outcomes
P = total
number of possible outcomes
P(number more than 4) = 2 numbers more than 4
6 possible outcomes
= 2
6
= 1
3
 0.33  33%
The theoretical probability of rolling a number more than 4 is
1 , 0.33, or 33%.
3
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10-4 Theoretical Probability
Additional Example 2A: School Application
The coach has written the names of each of
the track team members on an index card. She
draws randomly from these cards to choose a
student to run a sprint and then replaces the
card in the stack.
A. If there are 13 boys and 10 girls on the
team, what is the theoretical probability
that a girl’s name will be drawn?
Find the
number
of
girls
on
the
team
P(girl) =
theoretical
number of members on the team
probability.
10
Substitute.
=
23
Course 2
10-4 Theoretical Probability
Additional Example 2B: School Application
The coach has written the names of each of
the track members on an index card. She
draws randomly from these cards to choose a
student to run a sprint and then replaces the
card in the stack.
B. If there are 13 boys and 10 girls on the team,
what is the theoretical probability that a boy’s
name will be drawn?
Find the
number
of
boys
on
the
team
P(boy) =
theoretical
number of members on the team
probability.
= 13
Substitute.
23
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10-4 Insert
Lesson
Title Here
Theoretical
Probability
Try This: Example 2A
A teacher has written the name of each student
on a piece of paper and placed the names in a
box. She randomly draws a paper from the box
to determine which student will present the
answer to the problem of the day.
A. If there are 15 boys and 12 girls in the
class, what is the theoretical probability
that a girl’s name will be drawn?
Find the
number
of
girls
in
the
class
P(girl) =
theoretical
number of students in the class
probability.
12
Substitute.
=
27
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10-4 Insert
Lesson
Title Here
Theoretical
Probability
Try This: Example 2B
A teacher has written the name of each student
on a piece of paper and placed the names in a
box. She randomly draws a paper from the box
to determine which student will present the
answer to the problem of the day.
B. If there are 15 boys and 12 girls in the class,
what is the theoretical probability that a boy’s
name will be drawn?
Find the
number
of
boys
in
the
class
P(boy) =
theoretical
number of students in the class
probability.
Substitute.
= 15
27
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10-4 Theoretical
Insert Lesson
Probability
Title Here
Lesson Quiz
Find the probabilities. Write your answer as a
fraction, as a decimal to the nearest hundredth,
and as a percent to the nearest whole percent.
You have 11 cards, each with one of the letters
from the word mathematics.
1. Find the probability of drawing an m from the pile
2 , 0.18, 18%
of shuffled cards. 11
2. Find the probability of drawing a vowel. 4 , 0.36, 36%
11
3. Find the probability of drawing a consonant.
7 , 0.64, 64%
11
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