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Theoretical
Probability
9.4
Pre-Algebra
Warm Up
1. If you roll a number cube, what are the
possible outcomes?
1, 2, 3, 4, 5, or 6
2. Add 3 + 1. 1
36
6
4
3. Add 1 + 2 . 5
2
36
9
Learn to estimate probability using
theoretical methods.
Vocabulary
theoretical probability
equally likely
fair
mutually exclusive
Theoretical probability is used to estimate
probabilities by making certain assumptions
about an experiment. Suppose a sample space
has 5 outcomes that are equally likely, that is,
they all have the same probability, x. The
probabilities must add to 1.
x+x+x+x+x=1
5x = 1
x=1
5
A coin, die, or other object is called fair if all
outcomes are equally likely.
Example: Calculating
Theoretical Probability
An experiment consists of spinning this
spinner once.
A. What is the probability of
spinning a 4?
The spinner is fair, so all 5 outcomes
are equally likely. The probability of
spinning a 4 is P(4) = 1 .
5
Example: Calculating
Theoretical Probability
An experiment consists of spinning this
spinner once.
B. What is the probability of
spinning an even number?
There are 2 outcomes in the
event of spinning an even
number: 2 and 4.
P(spinning an even number) = number of possible even numbers
5
2
=5
Example: Calculating
Theoretical Probability
An experiment consists of spinning this
spinner once.
C. What is the probability of
spinning a number less
than 4?
There are 3 outcomes in the
event of spinning a number
less than 4: 1, 2, and 3.
P(spinning a number less than 4)= 3
5
Try This
An experiment consists of spinning this
spinner once.
A. What is the probability of
spinning a 1?
The spinner is fair, so all 5 outcomes
are equally likely. The probability of
spinning a 1 is P(1) = 1 .
5
Try This
An experiment consists of spinning this
spinner once.
B. What is the probability of
spinning an odd number?
There are 3 outcomes in the
event of spinning an odd
number: 1, 3, and 5.
P(spinning an odd number) = number of possible odd numbers
5
3
=5
Try This
An experiment consists of spinning this
spinner once.
C. What is the probability of
spinning a number less than
3?
There are 2 outcomes in the
event of spinning a number less
than 3: 1 and 2.
2
P(spinning a number less than 3)= 5
Example: Calculating
Theoretical Probability for a
Fair Die and a Fair Coin
An experiment consists of rolling one fair die
and flipping a coin.
A. Show a sample space that has all outcomes
equally likely.
The outcome of rolling a 5 and flipping heads
can be written as the ordered pair (5, H).
There are 12 possible outcomes in the
sample space.
1H
2H
3H
4H
5H
6H
1T
2T
3T
4T
5T
6T
Example: Calculating
Theoretical Probability for a
Fair Die and a Fair Coin
An experiment consists of rolling one fair die
and flipping a coin.
B. What is the probability of getting tails?
There are 6 outcomes in the event “flipping tails”:
(1, T), (2, T), (3, T), (4, T), (5, T), and (6, T).
6
1
P(tails) =
=
12
2
Example: Calculating
Theoretical Probability for a
Fair Die and a Fair Coin
An experiment consists of rolling one fair die
and flipping a coin.
C. What is the probability of getting an even
number and heads?
There are 3 outcomes in the event “getting an
even number and heads”: (2, H), (4, H), (6, H).
3
1
P(even number and heads) =
=
12
4
Example: Calculating
Theoretical Probability for a
Fair Die and a Fair Coin
D. What is the probability of getting a prime
number?
There are 6 outcomes in the event “getting a
prime number”: (2, T), (3, T), (5, T), (2, H),
(3, H), (5, H).
6
1
P(prime number) =
=
12 2
Try This
An experiment consists of flipping two coins.
A. Show a sample space that has all
outcomes equally likely.
The outcome of flipping two heads can be written
as HH. There are 4 possible outcomes in the
sample space.
HH
TH
HT
TT
Try This Continued
An experiment consists of flipping two coins.
B. What is the probability of getting one head and
one tail?
There are 2 outcomes in the event “getting
one head and getting one tail”: (H, T) and
(T, H).
2
1
P(head and tail) = 4 = 2
Try This
An experiment consists of flipping two coins.
C. What is the probability of getting heads on
both coins?
There is 1 outcome in the event “both heads”:
(H, H).
1
P(both heads) =
4
Try This
An experiment consists of flipping two coins.
D. What is the probability of getting both tails?
There is 1 outcome in the event “both tails”:
(T, T).
1
P(both tails) =
4
Two events are mutually exclusive if they
cannot both occur in the same trial of an
experiment. Suppose both A and B are two
mutually exclusive events.
• P(both A and B will occur) = 0
• P(either A or B will occur) = P(A) + P(B)
Example: Find the Probability of
Mutually Exclusive Events
Suppose you are playing a game in which you roll
two fair dice. If you roll a total of five you will win.
If you roll a total of two, you will lose. If you roll
anything else, the game continues. What is the
probability that the game will end on your next
roll?
It is impossible to roll a total of 5 and a total of 2 at
the same time, so the events are mutually exclusive.
Add the probabilities to find the probability of the
game ending on your next roll.
Example Continued
The event “total = 5” consists of 4 outcomes, (1, 4),
4
(2, 3), (3, 2), and (4, 1), so P(total = 5) = . The
36
event “total = 2” consists of 1 outcome, (1, 1), so
P(total = 2) = 1 .
36
P(game ends) = P(total = 5) + P(total = 2)
4
1
= 36 + 36
= 5
36
5
The probability that the game will end is 36 , or about
13.9%.
Try This
Suppose you are playing a game in which you flip
two coins. If you flip both heads you win and if
you flip both tails you lose. If you flip anything
else, the game continues. What is the probability
that the game will end on your next flip?
It is impossible to flip both heads and tails at the
same time, so the events are mutually exclusive.
Add the probabilities to find the probability of the
game ending on your next flip.
Try This Continued
The event “both heads” consists of 1 outcome, (H, H),
so P(both heads) =1 . The event “both tails” consists of
4
1
1 outcome, (T, T), so P(both tails) = .
4
P(game ends) = P(both tails) + P(both heads)
=1+ 1
4 4
1
=2
1
The probability that the game will end is 2 , or 50%.
Lesson Quiz
An experiment consists of rolling a fair die.
Find each probability.
1
1. P(rolling an odd number)
21
2. P(rolling a prime number)
2
An experiment consists of rolling two fair
dice. Find each probability.
1
3. P(rolling two 3’s) 36
4. P(total shown > 10) 1
12