Basic Concepts of Probability and CountingSection 3.1

Summer 2013 - Math 1040

June 17

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Roadmap

Basic Concepts of Probability and CountingPages 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 possibleoutcomes of a probability experiment. Counting these events allow us tofind the probability of an event.

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Sample Spaces

A sample space develops by listing all possible results from a randomexperiment.

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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Sample Spaces

A sample space develops by listing all possible results from a randomexperiment.

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.

{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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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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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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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.

(1040) M 1040 - 3.1 June 17 4 / 12

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.

(1040) M 1040 - 3.1 June 17 4 / 12

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 thepossible number of outcomes is found by multiplying the number ofoutcomes in each sample space.

Example Rolling a 4-sided die and flipping a coin’s sample space has4 · 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 thefirst event with the number of ways to do the next event.

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Fundamental Counting Principle

If we combine two (or more) basic types of experiments, counting thepossible number of outcomes is found by multiplying the number ofoutcomes in each sample space.

Example Rolling a 4-sided die and flipping a coin’s sample space has4 · 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 thefirst 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 differentdesserts. If a meal comes with a main dish and a dessert, how manydifferent means can be made?

Answer 4 · 3 = 12 many meals.Example How many 4-character liceanse plates can be made from 26letters 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 twoevents can occur in sequence is m · n, the product of the number of waysm the first and the number of ways n the second can occur. This extendsto more than two events.

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Fundamental Counting Principle

Example A restaurant offers four different main dishes and 3 differentdesserts. If a meal comes with a main dish and a dessert, how manydifferent means can be made?Answer 4 · 3 = 12 many meals.

Example How many 4-character liceanse plates can be made from 26letters 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 twoevents can occur in sequence is m · n, the product of the number of waysm the first and the number of ways n the second can occur. This extendsto more than two events.

(1040) M 1040 - 3.1 June 17 6 / 12

Fundamental Counting Principle

Example A restaurant offers four different main dishes and 3 differentdesserts. If a meal comes with a main dish and a dessert, how manydifferent means can be made?Answer 4 · 3 = 12 many meals.Example How many 4-character liceanse plates can be made from 26letters 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 twoevents can occur in sequence is m · n, the product of the number of waysm the first and the number of ways n the second can occur. This extendsto more than two events.

(1040) M 1040 - 3.1 June 17 6 / 12

Fundamental Counting Principle

Example A restaurant offers four different main dishes and 3 differentdesserts. If a meal comes with a main dish and a dessert, how manydifferent means can be made?Answer 4 · 3 = 12 many meals.Example How many 4-character liceanse plates can be made from 26letters 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 twoevents can occur in sequence is m · n, the product of the number of waysm the first and the number of ways n the second can occur. This extendsto more than two events.

(1040) M 1040 - 3.1 June 17 6 / 12

Fundamental Counting Principle

Example A restaurant offers four different main dishes and 3 differentdesserts. If a meal comes with a main dish and a dessert, how manydifferent means can be made?Answer 4 · 3 = 12 many meals.Example How many 4-character liceanse plates can be made from 26letters 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 twoevents can occur in sequence is m · n, the product of the number of waysm the first and the number of ways n the second can occur. This extendsto more than two events.

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Classical / Theoretical Probability

The probability an event E will occur is denoted P(E ) and said, ”theprobability of event E .”

Classical or theoretical probability is used when each outcome in asample space is equally likely to occur. The probability of an event E isthen

P(E ) =Number of outcomes in E

Total outcomes in the sample space

Example For a coin flip, the sample space is {H,T}. The event E : coinflip results in a heads is 1

2 .

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Classical / Theoretical Probability

The probability an event E will occur is denoted P(E ) and said, ”theprobability of event E .”

Classical or theoretical probability is used when each outcome in asample space is equally likely to occur. The probability of an event E isthen

P(E ) =Number of outcomes in E

Total outcomes in the sample space

Example For a coin flip, the sample space is {H,T}. The event E : coinflip results in a heads is 1

2 .

(1040) M 1040 - 3.1 June 17 7 / 12

Classical / Theoretical Probability

Example A card is drawn from a standard deck of playing cards. What isthe probability that the card drawn is a heart?

P(E ) =13

52=

1

4= 0.25.

What is the probability the card is a face card? (A jack, queen, king, orace)

There are four suits (heart, diamond, club, spade) and four face cards.

P(E ) =4 · 452

=16

52≈ 0.3077.

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Classical / Theoretical Probability

Example A card is drawn from a standard deck of playing cards. What isthe probability that the card drawn is a heart?

P(E ) =13

52=

1

4= 0.25.

What is the probability the card is a face card? (A jack, queen, king, orace)

There are four suits (heart, diamond, club, spade) and four face cards.

P(E ) =4 · 452

=16

52≈ 0.3077.

(1040) M 1040 - 3.1 June 17 8 / 12

Classical / Theoretical Probability

Example A card is drawn from a standard deck of playing cards. What isthe probability that the card drawn is a heart?

P(E ) =13

52=

1

4= 0.25.

What is the probability the card is a face card? (A jack, queen, king, orace)

There are four suits (heart, diamond, club, spade) and four face cards.

P(E ) =4 · 452

=16

52≈ 0.3077.

(1040) M 1040 - 3.1 June 17 8 / 12

Classical / Theoretical Probability

Example A card is drawn from a standard deck of playing cards. What isthe probability that the card drawn is a heart?

P(E ) =13

52=

1

4= 0.25.

What is the probability the card is a face card? (A jack, queen, king, orace)

There are four suits (heart, diamond, club, spade) and four face cards.

P(E ) =4 · 452

=16

52≈ 0.3077.

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Empirical / Statistical Probability

Empirical or statistical probabilities are based on observations. These arealways relative frequencies.

P(E ) =f

n=

Frequency of the event

Frequency total

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Classical / Theoretical Probability

Example Here is the toy dog breed data from the American KennelSociety (registered number of dogs in thousands)

Chihuahua 23Maltese 13Pomeranian 18Poodle 30Pug 20Shih Tzu 27Yorkshire Terrier 48

Σf = 179

What is the probability the next dog registered is a poodle?

P(E ) = 30179 ≈ 0.1676.

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Classical / Theoretical Probability

Example Here is the toy dog breed data from the American KennelSociety (registered number of dogs in thousands)

Chihuahua 23Maltese 13Pomeranian 18Poodle 30Pug 20Shih Tzu 27Yorkshire Terrier 48

Σf = 179

What is the probability the next dog registered is a poodle?P(E ) = 30

179 ≈ 0.1676.

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Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?Let E be ’the card is a heart.’ P(E ′) = 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 ′) = 1− P(E ) ≈ 1− 0.3077 = 0.6923

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Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?

Let E be ’the card is a heart.’ P(E ′) = 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 ′) = 1− P(E ) ≈ 1− 0.3077 = 0.6923

(1040) M 1040 - 3.1 June 17 11 / 12

Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?Let E be ’the card is a heart.’

P(E ′) = 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 ′) = 1− P(E ) ≈ 1− 0.3077 = 0.6923

(1040) M 1040 - 3.1 June 17 11 / 12

Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?Let E be ’the card is a heart.’ P(E ′) = 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 ′) = 1− P(E ) ≈ 1− 0.3077 = 0.6923

(1040) M 1040 - 3.1 June 17 11 / 12

Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?Let E be ’the card is a heart.’ P(E ′) = 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 ′) = 1− P(E ) ≈ 1− 0.3077 = 0.6923

(1040) M 1040 - 3.1 June 17 11 / 12

Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?Let E be ’the card is a heart.’ P(E ′) = 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 ′) = 1− P(E ) ≈ 1− 0.3077 = 0.6923

(1040) M 1040 - 3.1 June 17 11 / 12

Complementary Events

Because probabiity must be a number between 0 and 1, we can use thisfact to find the probabiliy of the complement of E , or all the events notin E . This is done by

P(E ′) = 1− P(E )

Example What is the probability that a card drawn from a standard deckis not a heart?Let E be ’the card is a heart.’ P(E ′) = 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 ′) = 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 thefundamental counting principle and the complement rule.

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