Download 6.2 Basics of Probability

6.2 Basics of Probability
LEARNING GOAL
Know how to find probabilities using theoretical and
relative frequency methods and understand how to
construct basic probability distributions.
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6.2-1
Definitions
An event is a collection of one or more outcomes that
share a property of interest.
Outcomes are the most basic possible results of
observations or experiments.
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6.2-2
Expressing Probability
The probability of an event, expressed as
P(event), is always between 0 and 1 inclusive.
A probability of 0 means that the event is
impossible, and a probability of 1 means that
the event is certain.
Figure 6.2 The scale shows various degrees of certainty
as expressed by probabilities.
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6.2-3
Theoretical Probabilities
Theoretical Method for Equally Likely Outcomes
Step 1. Count the total number of possible outcomes.
Step 2. Among all the possible outcomes, count the
number of ways the event of interest, A, can occur.
Step 3. Determine the probability, P(A), from
number of ways A can occur
P(A) = total number of outcomes
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6.2-4
EXAMPLE 1 Guessing Birthdays
Suppose you select a person at random from a large
group at a conference. What is the probability that the
person selected has a birthday in July? Assume 365
days in a year.
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6.2-5
Counting Outcomes
Suppose process A has a possible outcomes and
process B has b possible outcomes. Assuming the
outcomes of the processes do not affect each other, the
number of different outcomes for the two processes
combined is a × b.
This idea extends to any number of processes.
For example, if a third process C has c possible
outcomes, the number of possible outcomes for the
three processes combined is a × b × c.
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6.2-6
EXAMPLE 2 Some Counting
a. How many outcomes are there if you roll a fair die
and toss a fair coin?
b.What is the probability of rolling two 1’s when two
fair dice are rolled?
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6.2-7
EXAMPLE 3 Counting Children
What is the probability that, in a randomly selected
family with three children, the oldest child is a boy, the
second child is a girl, and the youngest child is a girl?
Assume boys and girls are equally likely.
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6.2-8
Theoretical Probabilities
Relative Frequency Probabilities
The second way to determine probabilities is to
approximate the probability of an event A by making
many observations and counting the number of times
event A occurs.
This approach is called the relative frequency (or
empirical) method.
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6.2-9
Slide
6.2- 9
Relative Frequency Method
Step 1. Repeat or observe a process many times and
count the number of times the event of interest,
A, occurs.
Step 2. Estimate P(A) by
number of times A occurred
P(A) =
total number of observations
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6.2-10
Slide
6.2- 10
EXAMPLE 4 500-Year Flood
Geological records indicate that a river has crested above a
particular high flood level four times in the past 2,000 years.
What is the relative frequency probability that the river will crest
above the high flood level next year?
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6.2-11
Slide
6.2- 11
EXAMPLE 5 Which Method?
Identify the method that resulted in the following statements.
a. The chance that you’ll get married in the next year is zero.
b. Based on government data, the chance of dying in an
automobile accident is 1 in 7,000 (per year).
c. The chance of rolling a 7 with a twelve-sided die is 1/12.
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6.2-12
Slide
6.2- 12
Probability of an Event Not Occurring
Probability of an Event Not Occurring
If the probability of an event A is P(A), then the
probability that event A does not occur is P(not A).
Because the event must either occur or not occur, we
can write
P(A) + P(not A) = 1
or
P(not A) = 1 – P(A)
Note: The event not A is called the complement of the
event A; the “not” is often designated by a bar, so Ā
means not A.
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6.2-13
Slide
6.2- 13
EXAMPLE 8 Tossing Three Coins
Make a probability distribution for the number of heads that
occurs when three coins are tossed simultaneously.
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6.2-14
Slide
6.2- 14
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6.2-15