42
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

MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

Embed Size (px)

Citation preview

Page 1: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

MATH 105: Finite Mathematics8-2: The Binomial Probablity Model

Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Page 2: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

Page 3: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

Page 4: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

A Motivating Example

Some probability problems involve repeating the same experimentseveral times. For example, flipping a coin.

Example

An unfair coin with Pr[H] = 25 is flipped two times. Find the

probability of exactly one Heads.

Example

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

Example

The same unfiar coin as in the previous examples is flipped fourtimes. Find the probability of exactly one Heads.

Page 5: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

A Motivating Example

Some probability problems involve repeating the same experimentseveral times. For example, flipping a coin.

Example

An unfair coin with Pr[H] = 25 is flipped two times. Find the

probability of exactly one Heads.

Example

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

Example

The same unfiar coin as in the previous examples is flipped fourtimes. Find the probability of exactly one Heads.

Page 6: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

A Motivating Example

Some probability problems involve repeating the same experimentseveral times. For example, flipping a coin.

Example

An unfair coin with Pr[H] = 25 is flipped two times. Find the

probability of exactly one Heads.

Example

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

Example

The same unfiar coin as in the previous examples is flipped fourtimes. Find the probability of exactly one Heads.

Page 7: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

A Motivating Example

Some probability problems involve repeating the same experimentseveral times. For example, flipping a coin.

Example

An unfair coin with Pr[H] = 25 is flipped two times. Find the

probability of exactly one Heads.

Example

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

Example

The same unfiar coin as in the previous examples is flipped fourtimes. Find the probability of exactly one Heads.

Page 8: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 theprobability of getting exactly two heads. What about gettingexactly three heads? Exactly four heads?

Note:

A pattern emerges when we repeat the same action, flipping thecoin, multiple times. The Bernoulli Probability Formula gives us away to quickly compute such probabilities.

Page 9: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 theprobability of getting exactly two heads. What about gettingexactly three heads? Exactly four heads?

Note:

A pattern emerges when we repeat the same action, flipping thecoin, multiple times. The Bernoulli Probability Formula gives us away to quickly compute such probabilities.

Page 10: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

Page 11: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 torepeat 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 ofa success is p yields the probability formula:

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

Page 12: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 torepeat 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 ofa success is p yields the probability formula:

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

Page 13: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 torepeat 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 ofa success is p yields the probability formula:

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

Page 14: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 torepeat 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 ofa success is p yields the probability formula:

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

Page 15: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 torepeat 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 ofa success is p yields the probability formula:

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

Page 16: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 torepeat 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 ofa success is p yields the probability formula:

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

Page 17: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

Page 18: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 ] = 13

C (4, 2)

(1

3

)3 (2

3

)1

= 4

(1

27

) (2

3

)= 4

(2

81

)=

8

81

Page 19: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 ] = 13

C (4, 2)

(1

3

)3 (2

3

)1

= 4

(1

27

) (2

3

)= 4

(2

81

)=

8

81

Page 20: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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).

Page 21: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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).

Page 22: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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).

Page 23: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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).

Page 24: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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

≈ 0.0042

3 probability he fails (6 or less).

Page 25: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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).

Page 26: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 toeach 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

Page 27: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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-throwpercentage was 0.687. Suppose that shooting free-throws is aBernoulli process. If Duncan took 8 free-throws in a certain gamethat 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?

Page 28: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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-throwpercentage was 0.687. Suppose that shooting free-throws is aBernoulli process. If Duncan took 8 free-throws in a certain gamethat 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?

Page 29: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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-throwpercentage was 0.687. Suppose that shooting free-throws is aBernoulli process. If Duncan took 8 free-throws in a certain gamethat 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?

Page 30: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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-throwpercentage was 0.687. Suppose that shooting free-throws is aBernoulli process. If Duncan took 8 free-throws in a certain gamethat 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?

Page 31: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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-throwpercentage was 0.687. Suppose that shooting free-throws is aBernoulli process. If Duncan took 8 free-throws in a certain gamethat 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?

Page 32: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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-throwpercentage was 0.687. Suppose that shooting free-throws is aBernoulli process. If Duncan took 8 free-throws in a certain gamethat 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?

Page 33: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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

Page 34: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 35: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 36: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 37: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 38: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 39: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 40: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 repeated2 only two possible outcomes for each trial3 trials are independent of each other4 probability of a sucess does not change

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

Page 41: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 valueis answering the question: “if you roll a die many times, over thelong run, what will the average value be?”

For next time

Read section 8-3

Page 42: MATH 105: Finite Mathematics 8-2: The Binomial …math.wallawalla.edu/~duncjo/courses/math105/winter06/slides/finite...Introduction to Bernoulli Processes Bernoulli Trials and the

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 valueis answering the question: “if you roll a die many times, over thelong run, what will the average value be?”

For next time

Read section 8-3