Theoretical
Probability
Goal: to find the probability
of an event using
theoretical probability.
Probability
A
probability experiment is an activity in
which results are observed. Each
observation is called a trial, and each
result is called an outcome. The sample
space is the set of all possible outcomes
of an experiment.
An Event…
 An
event is any set of one or more
outcomes.
 The probability of an event, written
P(event), is a number from 0 (or 0%) to 1
(or 100%) that tells you how likely the
event is to happen.



A probability of 0 means the event is
impossible, or can never happen.
A probability of 1 means the event is
certain, or has to happen.
The probabilities of all possible outcomes in
the sample space add up to 1.
Theoretical probability
 Theoretical
probability is used to estimate
probabilities by making certain
assumptions about an experiment.
 The assumption is that all outcomes that
are equally likely, that is, they all have the
same probability.
 To find theoretical probability:
number of outcomes in the event
total possible outcomes
Example #1:
number of outcomes in the event
total possible outcomes
 An
experiment consists of rolling a fair die.
There are 6 possible outcomes: 1, 2, 3, 4, 5,
and 6.
 What is the probability of rolling a 3, P(3)?
 What
is the probability of rolling an odd
number, P(odd number)?
 What
is the probability of rolling a number
less than 5, P(less than 5)?
Example #2:
 An
number of outcomes in the event
total possible outcomes
experiment consists of rolling one fair
die and flipping a coin.
 Show a sample space that has all
outcomes equally likely.
1H
1T
2H
2T
3H
3T
4H
4T
5H
5T
6H
6T
 What
is the probability of getting tails,
P(tails)?
 What
is the probability of getting an even
number and heads, P(even # and
heads)?
 What
is the probability of getting a prime
number, P(prime)?
Example #3:
 Using
the spinner, find the
probability of each event.
P(spinning A)
P(spinning
C)
P(spinning
D)
Example #4:
 Suppose
you roll two fair
dice and are considering
the sum shown.
 Since each die represents a
unique number, there are
36 total outcomes.
 What is the probability of
rolling doubles?
 There are 6 outcomes of
rolling doubles: (1, 1), (2, 2),
(3, 3), (4, 4), (5, 5), and (6, 6).
 P(double) =
Example #4 cont….
 What
is the probability of
rolling a total of 10?
 There are 6 outcomes in the
event “a total of 10”: (4, 6),
(5, 5), and (6, 4).
 P(total = 10) =
 What is the probability that
the sum shown is less than
5?
 There are 6 outcomes in the
event “total < 5” :(1, 1), (1,
2), (1, 3), (2, 1), (2, 2), and (3,
1).
 P(total < 5) =
Example #5:
An experiment consists of rolling a
single fair die.
 Find
P(rolling an even number)
 Find
P(rolling a 3 or a 5)
Example #6:
Mrs. Rife has candy hearts in a jar. She
has 27 red hearts, 16 pink hearts, and
17 white hearts.
 What is the theoretical probability of
drawing a white heart?
A red heart?.
Example #7:
Find each probability.
 P(white)
 P(red)
 P(green)
 P(black)
 What
should be the sum of the probabilities? Is
it true?
Example #8:
Three fair coins are tossed: a penny, a dime,
and a quarter. Copy the table and then find
the following
Penny Dime Quarter Outcome
H
H
H
HHH
 P(HHT)
H
H
T
HHT
 P(TTT)
H
T
H
HTH
H
T
T
HTT
 P(0 tails)
T
H
H
THH
 P(1 tail)
T
H
T
THT
T
T
H
TTH
 P(2 heads)
T
T
T
TTT
 P(all the same)
Example #7:
Find each probability.
 P(white)
 P(red)
 P(green)
 P(black)
 What
should be the sum of the probabilities? Is
it true?