MATH 105: Finite Mathematics 8-2: The Binomial Probablity Model - - PowerPoint PPT Presentation

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MATH 105: Finite Mathematics 8-2: The Binomial Probablity Model - - PowerPoint PPT Presentation

Oct 13, 2022 179 likes •612 views

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion MATH 105: Finite Mathematics 8-2: The Binomial Probablity Model Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

MATH 105: Finite Mathematics 8-2: The Binomial Probablity Model

  • Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Outline

1

Introduction to Bernoulli Processes

2

Bernoulli Trials and the Bernoulli Probability Formula

3

Examples

4

Conclusion

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Outline

1

Introduction to Bernoulli Processes

2

Bernoulli Trials and the Bernoulli Probability Formula

3

Examples

4

Conclusion

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

A Motivating Example

Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[H] = 2

5 is flipped two times. Find the

probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three

  • times. Find the probability of exactly one Heads.

Example The same unfiar coin as in the previous examples is flipped four

  • times. Find the probability of exactly one Heads.
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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

A Motivating Example

Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[H] = 2

5 is flipped two times. Find the

probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three

  • times. Find the probability of exactly one Heads.

Example The same unfiar coin as in the previous examples is flipped four

  • times. Find the probability of exactly one Heads.
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SLIDE 6

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

A Motivating Example

Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[H] = 2

5 is flipped two times. Find the

probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three

  • times. Find the probability of exactly one Heads.

Example The same unfiar coin as in the previous examples is flipped four

  • times. Find the probability of exactly one Heads.
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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

A Motivating Example

Some probability problems involve repeating the same experiment several times. For example, flipping a coin. Example An unfair coin with Pr[H] = 2

5 is flipped two times. Find the

probability of exactly one Heads. Example The same unfair coin as in the previous example is flipped three

  • times. Find the probability of exactly one Heads.

Example The same unfiar coin as in the previous examples is flipped four

  • times. Find the probability of exactly one Heads.
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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Making the Process More Complicated

Example Now suppose that you flip the coin four times and wish to find the probability of getting exactly two heads. What about getting exactly three heads? Exactly four heads? Note: A pattern emerges when we repeat the same action, flipping the coin, multiple times. The Bernoulli Probability Formula gives us a way to quickly compute such probabilities.

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Making the Process More Complicated

Example Now suppose that you flip the coin four times and wish to find the probability of getting exactly two heads. What about getting exactly three heads? Exactly four heads? Note: A pattern emerges when we repeat the same action, flipping the coin, multiple times. The Bernoulli Probability Formula gives us a way to quickly compute such probabilities.

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Outline

1

Introduction to Bernoulli Processes

2

Bernoulli Trials and the Bernoulli Probability Formula

3

Examples

4

Conclusion

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Bernoulli Trials

In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Bernoulli Trials

In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Bernoulli Trials

In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Bernoulli Trials

In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Bernoulli Trials

In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Bernoulli Trials

In order to apply the Bernoulli Probability Formula, we need to repeat a certain type of action multiple times. Bernoulli Trial A Bernoulli Trial is an action which: There are only two possible outcomes (success and failure). The action is independent of previous results. The probability of a success is constant. Bernoulli Process A Bernoulli Process if n Bernoulli Trials in which the probability of a success is p yields the probability formula: Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Outline

1

Introduction to Bernoulli Processes

2

Bernoulli Trials and the Bernoulli Probability Formula

3

Examples

4

Conclusion

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Example Find the probability of 3 successes in 4 trials with Pr[ success ] = 1

3

C(4, 2) 1 3 3 2 3 1 = 4 1 27 2 3

  • = 4

2 81

  • = 8

81

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Example Find the probability of 3 successes in 4 trials with Pr[ success ] = 1

3

C(4, 2) 1 3 3 2 3 1 = 4 1 27 2 3

  • = 4

2 81

  • = 8

81

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10. 2 probablility he scores 8 or better. 3 probability he fails (6 or less).

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10. 2 probablility he scores 8 or better. 3 probability he fails (6 or less).

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10.

C(10, 7) 1 4 7 3 4 3 ≈ 0.003

2 probablility he scores 8 or better. 3 probability he fails (6 or less).

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10.

≈ 0.003

2 probablility he scores 8 or better. 3 probability he fails (6 or less).

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10.

≈ 0.003

2 probablility he scores 8 or better.

C(10, 8) 1 4 8 3 4 2 + C(10, 9) 1 4 9 3 4 1 + C(10, 10) 1 4 10 3 4 ≈ 0.0042

3 probability he fails (6 or less).

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10.

≈ 0.003

2 probablility he scores 8 or better.

≈ 0.00042

3 probability he fails (6 or less).

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Taking a Quiz

Example A student takes a multiple choice quiz with 4 possible answers to each of the 10 questions. If he guesses randomly, find the:

1 probability he scores 7 out of 10.

≈ 0.003

2 probablility he scores 8 or better.

≈ 0.00042

3 probability he fails (6 or less).

1 − Pr[ 7 or better ] = 1 −

  • C(10, 7)

1 4 7 3 4 3 + 0.0042

  • = 1 − [0.003 + 0.00042]

= 1 − 0.00342 ≈ 0.9966

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Shooting Free-throws

Example In the 1995-96 season at Virgina, Tim Duncan’s free-throw percentage was 0.687. Suppose that shooting free-throws is a Bernoulli process. If Duncan took 8 free-throws in a certain game that year, what is the probability that he:

1 makes all 8? 2 makes a majority of them? 3 Do you think that shooting free-throws are Bernoulli trials?

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Shooting Free-throws

Example In the 1995-96 season at Virgina, Tim Duncan’s free-throw percentage was 0.687. Suppose that shooting free-throws is a Bernoulli process. If Duncan took 8 free-throws in a certain game that year, what is the probability that he:

1 makes all 8? 2 makes a majority of them? 3 Do you think that shooting free-throws are Bernoulli trials?

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Shooting Free-throws

Example In the 1995-96 season at Virgina, Tim Duncan’s free-throw percentage was 0.687. Suppose that shooting free-throws is a Bernoulli process. If Duncan took 8 free-throws in a certain game that year, what is the probability that he:

1 makes all 8?

C(8, 8)(0.687)8(1 − 0.687)0) ≈ 0.04962

2 makes a majority of them? 3 Do you think that shooting free-throws are Bernoulli trials?

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Shooting Free-throws

Example In the 1995-96 season at Virgina, Tim Duncan’s free-throw percentage was 0.687. Suppose that shooting free-throws is a Bernoulli process. If Duncan took 8 free-throws in a certain game that year, what is the probability that he:

1 makes all 8?

≈ 0.04962

2 makes a majority of them? 3 Do you think that shooting free-throws are Bernoulli trials?

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Shooting Free-throws

Example In the 1995-96 season at Virgina, Tim Duncan’s free-throw percentage was 0.687. Suppose that shooting free-throws is a Bernoulli process. If Duncan took 8 free-throws in a certain game that year, what is the probability that he:

1 makes all 8?

≈ 0.04962

2 makes a majority of them?

≈ 0.782

3 Do you think that shooting free-throws are Bernoulli trials?

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Shooting Free-throws

Example In the 1995-96 season at Virgina, Tim Duncan’s free-throw percentage was 0.687. Suppose that shooting free-throws is a Bernoulli process. If Duncan took 8 free-throws in a certain game that year, what is the probability that he:

1 makes all 8?

≈ 0.04962

2 makes a majority of them?

≈ 0.782

3 Do you think that shooting free-throws are Bernoulli trials?

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Outline

1

Introduction to Bernoulli Processes

2

Bernoulli Trials and the Bernoulli Probability Formula

3

Examples

4

Conclusion

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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SLIDE 36

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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SLIDE 37

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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SLIDE 38

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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SLIDE 39

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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SLIDE 40

Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Important Concepts

Things to Remember from Section 8-2

1 Bernoulli processes require: 1

the same process is repeated

2

  • nly two possible outcomes for each trial

3

trials are independent of each other

4

probability of a sucess does not change

2 Pr[ r successes ] = C(n, r)(p)r(1 − p)n−r

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Next Time. . .

Next time we will cover our last section dealing with probability. This section covers expected value. An example of expected value is answering the question: “if you roll a die many times, over the long run, what will the average value be?” For next time Read section 8-3

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Introduction to Bernoulli Processes Bernoulli Trials and the Bernoulli Probability Formula Examples Conclusion

Next Time. . .

Next time we will cover our last section dealing with probability. This section covers expected value. An example of expected value is answering the question: “if you roll a die many times, over the long run, what will the average value be?” For next time Read section 8-3