MATH 105: Finite Mathematics 7-1: Sample Spaces...

Probability Sample Spaces Assigning Probability Conclusion

MATH 105: Finite Mathematics7-1: Sample Spaces and Assignment of Probability

Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Outline

1 Probability

2 Sample Spaces

3 Assigning Probability

4 Conclusion

Outline

1 Probability

2 Sample Spaces

4 Conclusion

Introduction to Probability

Many real world events can be considered chance or random. Theymay be deterministic, but we can not know or comprehend all thefactors which determine the outcome.

Example

You flip a coin. Air current, the arrangement of the coin on yourfinger, the force of your flip, and other factors all go together todetermine the outcome of Heads or Tails.

For any one toss, these factors are too complicated to take intoaccount, and the outcome appears random. Since the outcome isheads roughly half the time, we assign the following probabilities:

Pr [H] =1

2Pr [T ] =

Introduction to Probability

Many real world events can be considered chance or random. Theymay be deterministic, but we can not know or comprehend all thefactors which determine the outcome.

Example

You flip a coin. Air current, the arrangement of the coin on yourfinger, the force of your flip, and other factors all go together todetermine the outcome of Heads or Tails.

For any one toss, these factors are too complicated to take intoaccount, and the outcome appears random. Since the outcome isheads roughly half the time, we assign the following probabilities:

Pr [H] =1

2Pr [T ] =

Probability Vocabulary

Probability Terms

OutcomeA particular result of an activity or event.

EventA set of outcomes which share a common characteristic.

Sample SpaceThe set of all possible outcomes for an experiment. This isthe universal set for the experiment.

Equally Likely EventsAll events in the sample space have the same probability.

Probability Terms

Outline

1 Probability

2 Sample Spaces

4 Conclusion

Finding Sample Spaces

One of the first tasks in finding probability is to determine thesample space for the experiment.

Example

You flip a fair coin. What is the sample space for this experiment?

Example

You roll a six-sided die and note the number which appears on top.What is the sample space for this experiment?

Example

S = {H,T}

Example

S = {H,T}

Example

S = {H,T}

Example

S = {1, 2, 3, 4, 5, 6}

Finding More Sample Spaces

Example

You flip a coin and roll a die, and note the result of each. what isthe sample space for this experiment?

S = {H1,H2, . . ., H6,T1,T2, . . ., T6}

c(S) = 2 · 6 = 12

Example

S = {H1,H2, . . ., H6,T1,T2, . . ., T6}

c(S) = 2 · 6 = 12

Example

S = {H1,H2, . . ., H6,T1,T2, . . ., T6}

c(S) = 2 · 6 = 12

Different Sample Spaces for the Same Experiment

Example

You roll two dice and note both numbers. What is the samplespace for this experiment?

Example

You roll two dice and note the sum of the two numbers. What isthe sample space for this experiment?

Example

S = {(1, 1), (1, 2), . . ., (2, 1), (2, 2), . . .} c(S) = 6 · 6 = 36

Example

S = {(1, 1), (1, 2), . . ., (2, 1), (2, 2), . . .} c(S) = 6 · 6 = 36

Example

S = {(1, 1), (1, 2), . . ., (2, 1), (2, 2), . . .} c(S) = 6 · 6 = 36

Example

S = {2, 3, . . ., 12} c(S) = 11

Taking a Quiz

Example

You take a True/False quiz with three questions. If you treat thisquiz as an experiment, what is the sample space?

Taking a Quiz

Example

S = {TTT ,TTF , . . ., FFF} c(S) = 8

Taking a Quiz

Example

S = {TTT ,TTF , . . ., FFF} c(S) = 8

Now that we have some practice identifying sample spaces, it is timeto start assigning probabilities.

Outline

1 Probability

2 Sample Spaces

4 Conclusion

Taking a Quiz

Example

How likely are you to get all three answers in the True/False quizcorrect if you guess on each question?

Rules for Assigning Probability

For each outcome W , 0 ≤ Pr [W ] ≤ 1

The sum of the probabilities of all outcomes is one.

Equally Likely Outcomes

Pr [TTT ] = Pr [TTF ] = . . . = Pr [FFF ] =1

Taking a Quiz

Example

S = {TTT ,TTF ,TFT ,TFF ,FTT ,FTF ,FFT ,FFF}

Taking a Quiz

Example

A few rules before we actually assign probabilities.

Taking a Quiz

Example

Taking a Quiz

Example

Taking a Quiz

Example

Probability Model

When you find the sample space for an experiment and assignprobabilities to each element of the sample space, you areconstructing a probability model.

Example

A six sided die is weighted so that the 1 is twice as likely as anyother number and all other numbers are equally likely. Find theprobability model.

S = { 1, 2, 3, 4, 5, 6, }2x x x x x x

Pr [1] =2

7Pr [2] = Pr [3] = Pr [4] = Pr [5] = Pr [6] =

Probability Model

Example

S = { 1, 2, 3, 4, 5, 6, }2x x x x x x

Pr [1] =2

7Pr [2] = Pr [3] = Pr [4] = Pr [5] = Pr [6] =

Probability Model

Example

S = { 1, 2, 3, 4, 5, 6, }2x x x x x x

Pr [1] =2

7Pr [2] = Pr [3] = Pr [4] = Pr [5] = Pr [6] =

Probability Model

Example

S = { 1, 2, 3, 4, 5, 6, }2x x x x x x

Pr [1] =2

7Pr [2] = Pr [3] = Pr [4] = Pr [5] = Pr [6] =

Probabilities of Events

To find the probability of an event in sample spaces with equallylikely outcomes, we use the following probability formula.

Probability of an Event

If E is a subset of a sample space S in which all outcomes areequally likely, then

Pr [E ] =c(E )

Example

You guess on all 3 questions in the True/False quiz seen earlier.What is the probability that you miss one?

E = {TTF ,TFT ,FTT} Pr [E ] =c(E )

Pr [E ] =c(E )

Example

Pr [E ] =c(E )

Example

Pr [E ] =c(E )

Example

Drawing Balls from an Urn

Example

A jar contains 8 balls: 4 green, 3 blue, and 1 red. You pick oneball at random. Find:

1 The probability the ball you draw is green.

2 The probability the ball you draw is not red.

Example

An urn contains 3 balls: one red, one green, and one yellow. Youdraw the balls out one-by-one at random. What is the probabilitythat the yellow ball is not drawn drawn last?

Drawing Balls from an Urn

Example

A jar contains 8 balls: 4 green, 3 blue, and 1 red. You pick oneball at random. Find:

1 The probability the ball you draw is green.

2 The probability the ball you draw is not red.

Example

An urn contains 3 balls: one red, one green, and one yellow. Youdraw the balls out one-by-one at random. What is the probabilitythat the yellow ball is not drawn drawn last?

Rolling Two Dice

Example

You roll two fair six-sided dice and note the sum of the rolls. Findeach probability.

1 Pr [ sum is 7 ]

2 Pr [ sum is 4 ]

3 Pr [ sum is 4 or 7 ]

4 Pr [ sum is 4 and 7 ]

Outline

1 Probability

2 Sample Spaces

4 Conclusion

Important Concepts

Things to Remember from Section 7-1

1 Probability Vocabulary: Outcomes, Events, Sample Spaces

2 Finding Sample Spaces

3 Building Probability Models

4 Assigning Probabilities to Events

Important Concepts

Next Time. . .

Since probabilities are based on sets: the sample space and events,it is conceivable that tools used to work with sets would also beimportant in working with probabilities.

Indeed, next time we will use rules for combining sets and VennDiagrams to help solve probability problems.

For next time

Read Section 7-2 (pp 376-384)

Do Problem Sets 7-1 A,B

Next Time. . .

Since probabilities are based on sets: the sample space and events,it is conceivable that tools used to work with sets would also beimportant in working with probabilities.

Indeed, next time we will use rules for combining sets and VennDiagrams to help solve probability problems.

For next time

Read Section 7-2 (pp 376-384)

Do Problem Sets 7-1 A,B

MATH 105: Finite Mathematics 7-1: Sample Spaces...

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