BASIC CONCEPTS IN PROBABILITY
Section 5.1
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Objectives
1.
2.
3.
4.
Construct sample spaces
Compute and interpret probabilities
Approximate probabilities using the Empirical Method
Approximate probabilities by using simulation
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OBJECTIVE 1
Construct sample spaces
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Probability
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”.
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:
Describe the sample space for each of the following experiments:
a) The toss of a coin
•
b)
The roll of a die
•
c)
The sample space is {Heads, Tails}.
The sample space is {1, 2, 3, 4, 5, 6}
Selecting a student at random from a list of 10,000 at a large
university
•
The sample space consists of the 10,000 students
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Probability Model
We are often concerned with occurrences that consist of several
outcomes. For example, when rolling a die, we might be interested in
the probability of rolling an odd number. Rolling an odd number
corresponds to the collection of outcomes {1, 3, 5} from the sample
space {1, 2, 3, 4, 5, 6}. In general, a collection of outcomes of a sample
space is called an event.
Once we have a sample space, we need to specify a probability of
each event. This is done with a probability model. We use the letter
“𝑃” to denote probabilities. For example, we denote the probability that
a tossed coin lands heads by 𝑃(Heads).
In general, if A denotes an event, the probability of event A is denoted
by 𝑃(A).
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OBJECTIVE 2
Compute and interpret probabilities
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Probability Rules
The probability of an event is always between 0 and 1. In other words,
for any event A, 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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Probabilities with Equally Likely Outcomes
If a sample space has 𝑛 equally likely outcomes and an event A has 𝑘
outcomes, then
𝑃(A) =
Number of outcomes in A
𝑘
=𝑛
Number of outcomes in the sample space
Example:
In the Georgia Cash-4 Lottery, a winning number between 0000 and 9999
is chosen at random, with all the possible numbers being equally likely.
What is the probability that all four digits are the same?
Solution:
The outcomes in the sample space are the numbers from 0000 to 9999, so
there are 10,000 equally likely outcomes in the sample space. There are
10 outcomes for which all the digits are the same: 0000, 1111, 2222, and
so on up to 9999. The probability is:
10
𝑃(All four digits the same) = 10,000 = 0.001
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Example – Probabilities
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.
a) What is the probability that there are exactly two girls?
b) What is the probability that all three children are the same gender?
Solution:
a) Of the eight equally likely outcomes, the three outcomes BGG, GBG,
3
and GGB correspond to having two girls. Therefore, 𝑃(Two girls) = 8 .
b)
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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Sampling is a Probability Experiment
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 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.
a) What is the probability that the sampled family owns a house?
b) What is the probability that the sampled family rents?
Solution:
a) The sample space consists of 10,000 households. Of these, 4753 own a
4753
house. Therefore, 𝑃(Own a house) = 10,000 = 0.4753.
b)
The number of families who rent is 912 + 2857 = 3769. Therefore, 𝑃(Rents)
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. There are no hard-and-fast rules
as to just how small a probability needs to be before an event is
considered unusual, but we will use the following rule of thumb.
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 unusual?
Solution:
The event of choosing a math major consists of 150 students out of a
150
total of 5000. The probability of choosing a math major is
= 0.03.
5000
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.
Example:
In 2010, there were 2,046,935 boys and 1,952,451 girls born in the U.S.
Approximate the probability that a newborn baby is a boy.
Solution:
We compute the number of times the experiment has been repeated:
2,046,935 + 1,952,451 = 3,999,386 births.
The proportion of births that are boys is
𝑃(Boy) ≈ 0.5118.
2,046,935
3,999,386
= 0.5118. We approximate
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OBJECTIVE 4
Approximate probabilities by using simulation
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Simulation
In practice, it can be difficult or impossible to repeat an experiment many times
in order to approximate a probability with the Empirical Method. In some cases,
we can use technology to repeat an equivalent virtual experiment many times.
Conducting a virtual experiment in this way is called simulation.
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Simulation on the TI-84 PLUS
The randInt command on the TI-84 PLUS calculator may be used to
simulate the rolling of a single die. To access this command, press
MATH, scroll to the PRB menu, and select randInt. The following
screenshots illustrate how to simulate the rolling of a die 100 times.
The outcomes are stored in list L1.
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Simulation in EXCEL
Suppose that three dice are rolled, the smallest possible total is 3 and the largest possible total
is 18. Technology, such as EXCEL, can be used to “roll” the dice as many times as desired.
Following are the frequencies for each
possible outcome when the dice are
“rolled” 1000 times. We may, for example,
estimate the probability of obtaining the
128
number 10 as 𝑃(10) =
= 0.128.
1000
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You Should Know…
•
•
•
•
•
•
The Law of Large Numbers
The definition of:
• Probability
• Sample space
• Event
How to compute probabilities with equally likely outcomes
The rule-of-thumb for determining when an event A is unusual (if
𝑃(A) < 0.05)
How to approximate probabilities using the Empirical Method
How to approximate probabilities using simulation
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