Section 11.4
Fundamentals of
Probability
1.
Objectives
Compute theoretical probability.
2.
Compute empirical probability.
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Section 11.4
Probability
Probability is the measure of how likely it is
for an event to occur.
In math, probability takes on a value
between 0 and 1 (including the
endpoints).
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Section 11.4
2
Theoretical Probability – definitions
and background
Probability
Section 11.4
The closer the
probability of a
given event is to 1,
the more likely it is
that the event will
occur.
The closer the
probability of a
given event is to 0,
the less likely that
the event will occur.
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Theoretical Probability – definitions
and background
Experiment is any occurrence for which
the outcome is uncertain.
Sample space is the set of all possible
outcomes of an experiment, denoted by S.
Event, denoted by E is any subset of a
sample space.
The sum of the theoretical probabilities of
all possible outcomes is 1.
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Section 11.4
Example: In the experiment of 2 coin
tosses, it is possible to write sample
space as
S = {HH, HT, TH, TT}.
The event, E, of one head and one tail is
E = {HT, TH}
n(E) = 2
n(S) = 4
n(S) is the number of all possible outcomes
for an experiment.
Section 11.4
Theoretical Probability
n(E) is the number of outcomes for any
event.
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Computing Theoretical Probability
Computing Theoretical Probability
If an event E has n(E) equally likely
outcomes and its sample space S has n(S)
equally-likely outcomes, the theoretical
probability of event E, denoted by P(E),
is:
P(E) = number of outcomes in event E =
total number of possible outcomes
n( E )
n( S )
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Section 11.4
Back to our experiment of 2 coin tosses:
Since our sample space S contained 4
equally-likely outcomes, and the event, E,
of one head and one tail, contained 2
equally-likely outcomes, the probability
that the outcome of 2 coin tosses will be
one head and one tail is
2 1
= .
4 2
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Example 1 - Computing Theoretical
Probability
Example 1 - Computing Theoretical
Probability
Solution:
a) There is only one way to roll a 6 (E = {6}),
so n(E) = 1.
a)
A fair die is rolled once. Find the
probability of rolling
There are 6 possible, equally likely outcomes
when rolling one die (S = {1, 2, 3, 4, 5,
6}), so n(S) = 6
6
P(3) = number of outcomes that result in 3 =
total number of possible outcomes
b) an number less than 5
Note: When rolling a fair die, each of the 6
numbers is equally likely.
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Section 11.4
Example 1 - Computing Theoretical
Probability
b) E = {1, 2, 3, 4}
n(E) = 4
n(S) is still 6, so
P(E) =
4 2
= ≈ .67
6 3
9
n( E ) 1
=
n( S ) 6
The standard 52-card deck
Example 2 - Probability and a Deck
of 52 Cards
Example 2 - Probability and a Deck
of 52 Cards
a.
You are dealt one card from a standard 52card deck. Find the probability of being
dealt a(n)
ace
Solution:
P(ace) =
number of outcomes that result in a ace =
total number of possible outcomes
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n( ace) 4
=
=
n( S )
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Applies to situations in which we observe
how frequently an event occurs.
Computing Empirical Probability
The empirical probability of event E is:
P(E) = observed number of times E occurs =
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If one person is randomly selected from the
population described above, find the
probability that the person is married.
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Solution. The probability of selecting a
married is the observed number of
married people 124.9 (million), divided by
the total number of U.S. adults, 212.5
(million).
P (selecting a married) = number married_ =
total number of adults
124.9
212.5
≈ .59 (rounded to the nearest one-hundredth)
Section 11.4
Section 11.4
Populations in millions
Example 4 - Computing Empirical
Probability
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n( hearts) 13 1
=
=
n( S )
52 4
Example 4 - Computing Empirical
Probability
total number of observed occurrences
n( E )
n( S )
Heart
Solution:
P(heart)
= number of outcomes that result in a heart
total number of possible outcomes
=
Empirical Probability
b.
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