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

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

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

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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).

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).

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).

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).

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).

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).

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

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?

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?

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?

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?

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?

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?

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

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

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

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

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

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

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

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

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

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