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SECTION 4.1
BASIC IDEAS
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Objectives
1.
2.
3.
Construct sample spaces
Compute and interpret probabilities
Approximate probabilities using the Empirical Method
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Objective 1
Construct sample spaces
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Probability Experiment
A probability experiment is one in which we do not know
what any individual outcome will be, but we do know how a
long series of repetitions will come out.
For example, if we toss a fair coin,
we do not know what the outcome of
a single toss will be, but we do know
what the outcome of a long series
of tosses will be – about half “heads”
and half “tails”.
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Probability
The probability of an event is the proportion of times that
the event occurs in the long run.
So, for a “fair” coin, that is, one
that is equally likely to come up
heads as tails, the probability of
heads is 1/2 and the probability
of tails is 1/2.
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Law of Large Numbers
The law of large numbers says that as a probability
experiment is repeated again and again, the proportion of
times that a given event occurs will approach its probability.
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Sample Space
The collection of all the possible outcomes of a probability
experiment is called a sample space.
Example:
 Suppose that a coin is tossed. The sample space consists of:
{Heads, Tails}

Suppose that a standard die is rolled. The sample space
consists of:
{1, 2, 3, 4, 5, 6}
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Event
We are often concerned with occurrences that consist of several
outcomes. For example, when rolling a die, we might be
concerned with the possibility of rolling an odd number. A
collection of outcomes of a sample space is called an event.
Example:
A probability experiment consists of rolling a die. The sample
space is {1, 2, 3, 4, 5, 6}.
The event of rolling an odd number = {1, 3, 5}.
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Probability Model
Once we have a sample space for an experiment, we need to
specify the probability of each event. This is done with a
probability model. We use the letter “P” to denote probabilities.
For example, if we toss a coin, we denote the probability that the
coin lands heads by “P(Heads).”
Notation:
If A denotes an event, the probability of event A is denoted by
P(A).
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Probabilities With Equally Likely Outcomes
If a sample space has n equally likely outcomes, and an
event A has k outcomes, then
Number of outcomes in A
k
P(A) 

Number of outcomes in the sample space n
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Objective 2
Compute and interpret probabilities
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Example
A fair die is rolled. Find the probability that an odd number comes up.
Solution:
The sample space has six equally likely outcomes:
{1, 2, 3, 4, 5, 6}
The event of an odd number has three outcomes:
{1, 3, 5}
The probability is:
3 1
P(Odd Number)  
6 2
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Example
A family has three children. Denoting a boy by B and a girl by G, we can
denote the genders of these children from oldest to youngest. For example,
GBG means the oldest child is a girl, the middle child is a boy, and the
youngest child is a girl. There are eight possible outcomes: BBB, BBG, BGB,
BGG, GBB, GBG, GGB, and GGG. Assume these outcomes are equally likely.
What is the probability that all three children are the same gender?
Solution:
Of the eight equally likely outcomes, the two outcomes BBB and GGG
correspond to having all children of the same gender. Therefore
2 1
P(All three have same gender)  
8 4
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Probability Rules
The probability of an event is always between 0 and 1.
That is, 0 ≤ P(A) ≤ 1.
If A cannot occur, then P(A) = 0.
If A is certain to occur, then P(A) = 1.
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Sampling From a Population
Sampling an individual from a population is a probability experiment. The
population is the sample space and members of the population are equally
likely outcomes.
Example:
There are 10,000 families in a certain town categorized as follows:
Own a house
Own a condo
Rent a house
Rent an apartment
4753
1478
912
2857
A pollster samples a single family from this population. What is the
probability that the sampled family rents?
Solution:
The number of families who rent is 912 + 2857 = 3769. Therefore, the
probability that the sampled family rents is 3769/10,000 = 0.3769.
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Unusual Events
An unusual event is one that is not likely to happen. In other words, an event
whose probability is small.
A rule of thumb is that any event whose probability is less than 0.05 is
considered to be unusual.
Example:
In a college of 5000 students, 150 are math majors. A student is selected at
random and turns out to be a math major. Is this an unusual event?
Solution:
The event of choosing a math major consists of 150 students out of a total of
5000 students. The probability of choosing a math major is 150/5000 =
0.03. Since 0.03 < 0.05, this would be considered an unusual event.
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Objective 3
Approximate probabilities using the Empirical
Method
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Approximating Probabilities with the Empirical Method
The law of large numbers says that if we repeat a probability
experiment a large number of times, then the proportion of times
that a particular outcome occurs is likely to be close to the true
probability of the outcome.
The Empirical Method consists of repeating an experiment a
large number of times, and using the proportion of times an
outcome occurs to approximate the probability of the outcome.
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Example
The Centers for Disease Control reports that in the year 2002 there were
2,057,979 boys and 1,963,747 girls born in the U.S. Approximate the
probability that a newborn baby is a boy.
Solution:
The number of times that the experiment has been repeated is:
2,057,979 boys + 1,963,747 girls = 4,021,726 births
The proportion of births that are boys is:
2,057,979/4,021,726 = 0.5117
Therefore, the probability that a newborn baby is a boy is approximated by
0.5117.
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Do You Know…
•
•
•
•
How to construct a sample space?
How to compute probabilities of equally likely events?
The rules of probability?
How to compute probabilities using the Empirical Method?
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