Probability. An experiment is any process that allows researchers to obtain observations and which...

Probability

Probability

An experiment is any process that allows researchers to obtain observations and which leads to a single outcome which cannot be predicted with certainty

Probability

An experiment is any process that allows researchers to obtain observations and which leads to a single outcome which cannot be predicted with certainty

Example: A coin is tossed…possible outcomes are H, T.

Example: Two coins are tossed…possible outcomes are HH, HT, TH, TT.

Copyright © 2013 Pearson Education, Inc.. All rights reserved.

Tree diagram for the coin-tossing experiment

Probability

An outcome is also called a sample point.

The collection of all possible sample points is called the sample space.

.

Probability

An outcome is also called a sample point.

The collection of all possible sample points is called the sample space.

Example: Roll a die and observe the result. The sample space consists of the set

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

Example

When we roll two dice the sample space is all 36 possible pairs of rolls

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2,6

3,1 3,2 3,3 3,4 3,5 3,6

4,1 4,2 4,3 4,4 4,5 4,6

5,1 5,2 5,3 5,4 5,5 5,6

6,1 6,2 6,3 6,4 6,5 6,6

The probability of a single outcome

If an experiment is repeated many times, and if a particular outcome is known or observed to occur, for example, 1/6th of the time, we say that the probability of that outcome is 1/6.

Sometimes one can compute probabilities exactly. E.g., if a bowl contains 5 red chips and 3

Blue chips and if a chip is selected at random, then the probability a blue chip is selected is 3/8.

In general, it may be difficult to determine the probability of an outcome. (for example, what is the probability of rain tomorrow).

In general, the probabilty that a particular outcome occurs is defined to be the fraction of times which that outcome occurs in the long run.

Copyright © 2013 Pearson Education, Inc.. All rights reserved.

Proportion of heads in N tosses of a coin

Equi-probability sample spaces

A sample space S in which all sample points are equally likely to occur is called an equi-probability space.

Definition: For an equi-probability space, the probability of a particular outcome is 1/n, where n is the number of sample points in S.

Example

A coin is tossed and a die is rolled. What is the probability that the outcome is H6?

Example

A coin is tossed and a die is rolled. What is the probability that the outcome is H6?

Answer: S={H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Therefore P(H6) = 1/12.

Events

It turns out that we are usually interested NOT in a single outcome, but rather a collection of outcomes forming a subset of the sample space.

Definition: An event is a specific collection of sample points.

Sample Space

Sample Space

Event

Simple Events (all the red dots)

We often represent the sample space with a Venn Diagram.

Probability of an event

NOTE: True in general NOT just for equi-probability case

NOTE

For an equi-probability sample space, the probability of an event E occurring is simply the number of sample points in E divided by the total number of sample points in the sample space S.

Sample Space

Example: Roll two dice. What is the probability of rolling a 9?

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2,6

3,1 3,2 3,3 3,4 3,5 3,6

4,1 4,2 4,3 4,4 4,5 4,6

5,1 5,2 5,3 5,4 5,5 5,6

6,1 6,2 6,3 6,4 6,5 6,6

Sample Space


)3,6()4,5()5,4()6,3()9( PPPPP

Sample Space


9

1

36

4

36

1

36

1

36

1

36

1

)3,6()4,5()5,4()6,3()9(

PPPPP

Properties of Probability

1)()()()( (2)

then},,{

points sample of up made is space sample theIf

1)(0 (1)

satisfies spoint sample single a ofy Probabilit The

321

321

n

n

sPsPsPsP

ssss

sP

Union

The union of events A and B is the event that either A or B (or both) occur.

Notation:

U B

Union

The union of events A and B is the event that A or B (or both) occur.

AB

A or B

Intersection

The intersection of events A and B is the event that both A and B occur.

Notation:

Intersection

The intersection of events A and B is the event that both A and B occur.

A B

A and B

Compliment

The compliment of an event A is the event that A does not occur.

Notation:

Compliment


AC

Compliment Rule


We use AC to denote the compliment of A.

Addition Rule

P(A or B)= P(A) + P(B) - P(A and B)

“Proof”: Consider the case of an equi-probability space.

To find P(A or B) we must count all the points in the set

AUBWe can do that by adding together all of the points in A plus all the points in B. But this will count all of those points which lie in BOTH A and B twice so we must subtract the number of points in the intersection Letting N be the number of points in the entire sample space and using the notation that n(A) denotes the number of points in a set A, we obtain

)()()()()()()()()(

)( BAPBPAPN

BAn

N

Bn

N

An

N

BAnBnAnBAP

A I B.

ExampleFor an experiment of randomly selecting one

card from a deck of 52 cards, letA=event the card selected is the King of Hearts

B=event the card selected is a King

C=event the card selected is a Heart

D=event the card selected is a face card.

Find:

a) P(DC) b) P(B and C)

c) P(B or C) d) P(C and D)

e) P(A or B) f) P(B)

ExampleFor an experiment of randomly selecting one

card from a deck of 52 cards, letA=event the card selected is the King of Hearts

B=event the card selected is a King

C=event the card selected is a Heart

D=event the card selected is a face card.

Find:

a) P(DC) =40/52 b) P(B and C)= 1/52

c) P(B or C)=16/52 d) P(C and D)=3/52

e) P(A or B)=4/52 f) P(B)=4/52

Mutually Exclusive Events

Two events are mutually exclusive if

P (A and B) = 0.

In terms of sets, this means that the sets A and B to do not overlap.

For mutually exclusive sets, we have a simplified addition rule:

P(AUB) P(A) P(B)

Mutually Exclusive


P (A and B) = 0.

Suppose P (E) = .3, P (F) = .5, and E and F are mutually exclusive. Find:

P(E and F)= P(E or F)=

P(EC)= P(FC)=

P((E or F) C)= P((E and F) C)=

Mutually Exclusive


P (A and B) = 0.

Suppose P (E) = .3, P (F) = .5, and E and F are mutually exclusive. Find:

P(E and F) = 0 P(E or F) = 0.8

P(EC) = 0.7 P(FC) = 0.5

P((E or F) C)=0.2 P((E and F) C)=1

Example

A cage contains 7 black mice, 4 brown mice and 1 white mouse. A mouse is selected at random from the cage. What is the probability it is either a black mouse or a white mouse?

FrequencyBlack 7 Brown 4White 1Total 12

Example


Frequency ProbabilityBlack 7 7/12Brown 4 4/12White 1 1/12Total 12 1

Example


P (Black or White) =

Example


P (Black or White) = P(Black) +P(White)

Example


P (Black or White) = P(Black) +P(White)

= 7/12 + 1/12 = 8/12

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.a) Find the probability that a computer set purchased has one of the two problemsb) Neither c) Just a monitor problem

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.a) Find the probability that a computer set purchased has one of the two problems. (5.8%) b) Neitherc) Just a monitor problem

Example

In buying a new computer (tower, monitor, keyboard and mouse) studies show that 4% have problems with their mouse and 2% have problems with their monitor and 0.2% have problems with both before the expirations of their manufactured warranty.

a) Find the probability that a computer set purchased has one of the two problems. (5.8%)

b) Neither (94.2%)

c) Just a monitor problem (1.8%)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and

P(E or F)=0.6. Find:

P(E and F) P(EC or FC)

P(EC and FC) P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and


P(E and F) =0.1 P(EC or FC)


Example

Suppose P (E) = 0.4, P (F) = 0.3, and


P(E and F) =0.1 P(EC or FC)=0.9


Example

Suppose P (E) = 0.4, P (F) = 0.3, and



P(EC and FC)=0.4 P(EC and F)

Example

Suppose P (E) = 0.4, P (F) = 0.3, and



P(EC and FC)=0.4 P(EC and F)= 0.2

Review

Probabilities– Definitions of experiment, event, simple

event, sample space, probabilities, intersection, union compliment

– Finding Probabilities– Drawing Venn Diagrams – If A and B are two events then

P(A or B) = P(A) + P(B) - P(A and B),

P(not A) = 1 - P(A). – Two events A and B are mutually exclusive if

P(A and B) = 0.

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