10-4 Theoretical Probability
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
10-4 Theoretical Probability
I can find the theoretical probability of an event.
1. Two coins are tossed. What is the
probability of getting two heads? 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
10-4 Theoretical Probability
Problem of the Day
The name of a U.S. state is spelled out
with letter tiles. Then the tiles are placed
in a bag, and one is picked at random.
What state was spelled out if the
probability of picking the letter O is 1
?
2
3 ?1?
8 3
Ohio; Colorado; Oregon
10-4 Theoretical Probability
I can find the theoretical probability of an
event.
10-4 Theoretical Probability
Vocabulary
theoretical probability
equally likely
fair
10-4 Theoretical Probability
Theoretical probability is used to find the
probability of an event when all the outcomes are
equally likely.
Equally likely outcomes have the same
probability.
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.
10-4 Theoretical Probability
Additional Example 1A: Finding Theoretical
Probability
Andy has 20 marbles in a bag. Of these, 9 are clear
and 11 are blue. Find the probability of drawing a
clear marble from the bag? Write your answer as a
fraction, as a decimal, and as a percent.
number of ways the event can occur
P = total
number of equally likely 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
10-4 Theoretical Probability
Additional Example 1B: Finding Theoretical
Probability
Find the probability of drawing a blue marble from
the bag.
number of ways the event can occur
P = total
number of equally likely outcomes
of blue marbles Write the ratio.
P(blue) = number
total number of marbles
11
= 20
Substitute.
= 0.55 = 55%
Write as a decimal and write as a
percent.
The theoretical probability of drawing a clear marble is
11 , 0.55, or 55%.
20
10-4 Theoretical Probability
Check It Out: Example 1A
Jane has 20 marbles in a bag. Of these 8 are green.
Find the probability of drawing a green marble
from the bag? Write your answer as a fraction, as a
decimal, and as a percent.
number of ways the event can occur
P = total
number of equally likely 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
10-4 Theoretical Probability
Check It Out: Example 1B
Find 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.
P=
number of ways the event can occur
total number of equally likely 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
10-4 Theoretical Probability
Additional Example 2A: School Application
There are 13 boys and 10 girls on the track
team. The name of each of the team members
is written on an index card. A card is drawn at
random to choose a student to run a sprint
and the card is replaced in the stack.
Find the theoretical probability of drawing a
boy’s name.
P(boy) =
number of boys on the team
number of members on the team
P(boy)= 13
23
Substitute.
10-4 Theoretical Probability
Remember!
The sum of the probabilities of an event and its
complement is 1.
10-4 Theoretical Probability
Additional Example 2B: School Application
There are 13 boys and 10 girls on the track
team. The name of each of the team members
is written on an index card. A card is drawn at
random to choose a student to run a sprint
and the card is replaced in the stack.
Find the theoretical probability of drawing a
girl’s name.
P(boy) + P(girl) = 1
13
23 + P(girl) = 1
- 13
= - 13
23
23
10
P(girl) =
23
13
Substitute 23 for P(boy).
13
Subtract 23 from both sides
Simplify.
10-4 Theoretical Probability
Check It Out: 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.
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
10-4 Theoretical Probability
Check It Out: Example 2B
What is the theoretical probability that a boy’s
name will be drawn?
P(girl) + P(boy) = 1
12
27 + P(boy) = 1
- 12
= - 12
27
27
15
P(boy) =
27
12
Substitute 27 for P(girl).
12
Subtract 27 from both sides
Simplify.
10-4 Theoretical Probability
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
10-4 Theoretical Probability
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
of shuffled cards. 2 , 0.18, 18%
11
4 , 0.36, 36%
2. Find the probability of drawing a vowel. 11
3. Find the probability of drawing a consonant.
7 , 0.64, 64%
11
10-4 Theoretical Probability
Lesson Quiz for Student Response Systems
1. A number cube is rolled. Identify the probability
of getting a number less than 4 as a fraction, as a
decimal to the nearest hundredth, and as a percent
to the nearest whole percent.
A.
B.
C.
D.
10-4 Theoretical Probability
Lesson Quiz for Student Response Systems
2. There are 40 balls in a bag. Of these 8 are
white, 7 are blue, 12 are green, and 13 are yellow.
Identify the probability of drawing a white ball as a
fraction, as a decimal to the nearest hundredth,
and as a percent to the nearest whole percent.
A.
B.
C.
D.