Basic Concepts of Probability and Counting
Section 3.1
Summer 2013 - Math 1040
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Roadmap
Basic Concepts of Probability and Counting
Pages 128 - 137
I
Counting events, and The Fundamental Counting Principle
I
Theoretical probability and statistical probability
This section introduces the concept of a sample space, a list of all possible
outcomes of a probability experiment. Counting these events allow us to
find the probability of an event.
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Sample Spaces
A sample space develops by listing all possible results from a random
experiment.
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Sample Spaces
A sample space develops by listing all possible results from a random
experiment.
Example Rolling a 4-sided die’s sample space is {1, 2, 3, 4}.
Example A coin flip’s outcome is {H, T } for heads and tails.
Example Possible answer’s to, ”Do you want kids?” is a sample space:
{Yes, No, Maybe}.
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Events
Particular outcomes is called an event.
Example: We roll a 4-sided die. Here are some possible events:
I
You roll less than a 4.
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Events
Particular outcomes is called an event.
Example: We roll a 4-sided die. Here are some possible events:
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You roll less than a 4. {1, 2, 3} There are 3 ways.
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Events
Particular outcomes is called an event.
Example: We roll a 4-sided die. Here are some possible events:
I
You roll less than a 4. {1, 2, 3} There are 3 ways.
I
You roll an odd number.
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Events
Particular outcomes is called an event.
Example: We roll a 4-sided die. Here are some possible events:
I
You roll less than a 4. {1, 2, 3} There are 3 ways.
I
You roll an odd number. {1, 3} There are 2 ways.
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Events
Particular outcomes is called an event.
Example: We roll a 4-sided die. Here are some possible events:
I
You roll less than a 4. {1, 2, 3} There are 3 ways.
I
You roll an odd number. {1, 3} There are 2 ways.
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You roll a prime number.
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Events
Particular outcomes is called an event.
Example: We roll a 4-sided die. Here are some possible events:
I
You roll less than a 4. {1, 2, 3} There are 3 ways.
I
You roll an odd number. {1, 3} There are 2 ways.
I
You roll a prime number. {2, 3} There are 2 ways.
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Fundamental Counting Principle
If we combine two (or more) basic types of experiments, counting the
possible number of outcomes is found by multiplying the number of
outcomes in each sample space.
Example Rolling a 4-sided die and flipping a coin’s sample space has
4 · 2 = 8 outcomes:
{1H, 2H, 3H, 4H, 1T , 2T , 3T , 4T }
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Fundamental Counting Principle
If we combine two (or more) basic types of experiments, counting the
possible number of outcomes is found by multiplying the number of
outcomes in each sample space.
Example Rolling a 4-sided die and flipping a coin’s sample space has
4 · 2 = 8 outcomes:
{1H, 2H, 3H, 4H, 1T , 2T , 3T , 4T }
For an event, the rule is the same. Multiply the number of ways to do the
first event with the number of ways to do the next event.
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Fundamental Counting Principle
Example A restaurant offers four different main dishes and 3 different
desserts. If a meal comes with a main dish and a dessert, how many
different means can be made?
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Fundamental Counting Principle
Example A restaurant offers four different main dishes and 3 different
desserts. If a meal comes with a main dish and a dessert, how many
different means can be made?
Answer 4 · 3 = 12 many meals.
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Fundamental Counting Principle
Example A restaurant offers four different main dishes and 3 different
desserts. If a meal comes with a main dish and a dessert, how many
different means can be made?
Answer 4 · 3 = 12 many meals.
Example How many 4-character liceanse plates can be made from 26
letters and 10 digits (zero through nine)?
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Fundamental Counting Principle
Example A restaurant offers four different main dishes and 3 different
desserts. If a meal comes with a main dish and a dessert, how many
different means can be made?
Answer 4 · 3 = 12 many meals.
Example How many 4-character liceanse plates can be made from 26
letters and 10 digits (zero through nine)?
Answer There are 36 different characters each time.
36 · 36 · 36 · 36 = 364 = 1, 679, 616 many ways.
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Fundamental Counting Principle
Example A restaurant offers four different main dishes and 3 different
desserts. If a meal comes with a main dish and a dessert, how many
different means can be made?
Answer 4 · 3 = 12 many meals.
Example How many 4-character liceanse plates can be made from 26
letters and 10 digits (zero through nine)?
Answer There are 36 different characters each time.
36 · 36 · 36 · 36 = 364 = 1, 679, 616 many ways.
This is the fundamental counting principle: The number of ways two
events can occur in sequence is m · n, the product of the number of ways
m the first and the number of ways n the second can occur. This extends
to more than two events.
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Classical / Theoretical Probability
The probability an event E will occur is denoted P(E ) and said, ”the
probability of event E .”
Classical or theoretical probability is used when each outcome in a
sample space is equally likely to occur. The probability of an event E is
then
Number of outcomes in E
P(E ) =
Total outcomes in the sample space
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Classical / Theoretical Probability
The probability an event E will occur is denoted P(E ) and said, ”the
probability of event E .”
Classical or theoretical probability is used when each outcome in a
sample space is equally likely to occur. The probability of an event E is
then
Number of outcomes in E
P(E ) =
Total outcomes in the sample space
Example For a coin flip, the sample space is {H, T }. The event E : coin
flip results in a heads is 12 .
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Classical / Theoretical Probability
Example A card is drawn from a standard deck of playing cards. What is
the probability that the card drawn is a heart?
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Classical / Theoretical Probability
Example A card is drawn from a standard deck of playing cards. What is
the probability that the card drawn is a heart?
P(E ) =
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1
13
= = 0.25.
52
4
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Classical / Theoretical Probability
Example A card is drawn from a standard deck of playing cards. What is
the probability that the card drawn is a heart?
1
13
= = 0.25.
52
4
What is the probability the card is a face card? (A jack, queen, king, or
ace)
P(E ) =
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Classical / Theoretical Probability
Example A card is drawn from a standard deck of playing cards. What is
the probability that the card drawn is a heart?
1
13
= = 0.25.
52
4
What is the probability the card is a face card? (A jack, queen, king, or
ace)
P(E ) =
There are four suits (heart, diamond, club, spade) and four face cards.
P(E ) =
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4·4
16
=
≈ 0.3077.
52
52
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Empirical / Statistical Probability
Empirical or statistical probabilities are based on observations. These are
always relative frequencies.
P(E ) =
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f
Frequency of the event
=
n
Frequency total
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Classical / Theoretical Probability
Example Here is the toy dog breed data from the American Kennel
Society (registered number of dogs in thousands)
Chihuahua
Maltese
Pomeranian
Poodle
Pug
Shih Tzu
Yorkshire Terrier
Σf =
23
13
18
30
20
27
48
179
What is the probability the next dog registered is a poodle?
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Classical / Theoretical Probability
Example Here is the toy dog breed data from the American Kennel
Society (registered number of dogs in thousands)
Chihuahua
Maltese
Pomeranian
Poodle
Pug
Shih Tzu
Yorkshire Terrier
Σf =
23
13
18
30
20
27
48
179
What is the probability the next dog registered is a poodle?
30
P(E ) = 179
≈ 0.1676.
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
Example What is the probability that a card drawn from a standard deck
is not a heart?
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
Example What is the probability that a card drawn from a standard deck
is not a heart?
Let E be ’the card is a heart.’
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
Example What is the probability that a card drawn from a standard deck
is not a heart?
Let E be ’the card is a heart.’ P(E 0 ) = 1 − P(E ) = 1 − 0.25 = 0.75.
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
Example What is the probability that a card drawn from a standard deck
is not a heart?
Let E be ’the card is a heart.’ P(E 0 ) = 1 − P(E ) = 1 − 0.25 = 0.75.
What is the probabiliy that a card drawn is not a face card?
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
Example What is the probability that a card drawn from a standard deck
is not a heart?
Let E be ’the card is a heart.’ P(E 0 ) = 1 − P(E ) = 1 − 0.25 = 0.75.
What is the probabiliy that a card drawn is not a face card?
Let E be ’the card is a face card.’
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Complementary Events
Because probabiity must be a number between 0 and 1, we can use this
fact to find the probabiliy of the complement of E , or all the events not
in E . This is done by
P(E 0 ) = 1 − P(E )
Example What is the probability that a card drawn from a standard deck
is not a heart?
Let E be ’the card is a heart.’ P(E 0 ) = 1 − P(E ) = 1 − 0.25 = 0.75.
What is the probabiliy that a card drawn is not a face card?
Let E be ’the card is a face card.’
P(E 0 ) = 1 − P(E ) ≈ 1 − 0.3077 = 0.6923
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Assignments
Assignment:
1. Summarize this section.
2. Read pages 128 - 137
3. Page 138, 1 - 73 odd
4. Try It Yourself exercises 1, 3, 4, 5, 7, 9
Vocabulary: sample space, event, the fundamental counting principle,
theoretical probability, statistical probability, complementary events
Understand: Write out a list of all possilbe outcomes of an experiment.
This is the sample space. Count these events, and add up these events.
This way you can compute probabilites. Use techniques such as the
fundamental counting principle and the complement rule.
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