Probability

The Study of Randomness

The language of probability

Random in statistics does not mean “haphazard”.Random is a description of a kind of order that emerges only in the long run even though individual outcomes are uncertain.The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Probability Models

The sample space of a random event is the set of all possible outcomes.

What is the sample space for rolling a six-sided die?S = {1, 2, 3, 4, 5, 6}What is the sample space for flipping a coin and then choosing a vowel at random?

Tree diagram a

e

i

o

u

a

e

i

o

u

H

T

S={Ha, He, Hi, Ho, Hu, Ta, Te, Ti, To, Tu}

• What is the sample space for answering one true/false question?

• S = {T, F}

• What is the sample space for answering two true/false questions?

• S = {TT, TF, FT, FF}

• What is the sample space for three?

Tree diagram

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

True

False

True

True

False

False

True

True

True

True

False

False

False

False

Intuitive Probability

An event is an outcome or set of outcomes of a random phenomenon. An event is a subset of the sample space.For probability to be a mathematical model, we must assign proportions for all events and groups of events.

Basic Probability Rules

The probability P(A) of any event A satisfies 0 < P(A) < 1.Any probability is a number between 0 and 1, inclusive.If S is the sample space in a probability model, then P(S) = 1.All possible outcomes together must have probability of 1.

Complement Rule

The complement of any event A is the event that A does not occur, written as Ac. The complement rule states that

P(Ac) = 1 – P(A)

The probability that an event does not occur is 1 minus the probability that the event does occur.

Venn diagram: complement

A Ac

S

General Addition Rule for Unions of Two Events

For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B)

P(AB) = P(A) + P(B) – P(AB)

The simultaneous occurrence of two events is called a joint event.The union of any collections of event that at least one of the collection occurs.

Venn diagram: {A and B}

S

A

B

Venn diagram: disjoint events

(Mutually Exclusive)

A B

S

Addition Rule

Two events A and B are disjoint (also called Mutually Exclusive) if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint,

P(A or B) = P(A) + P(B) If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

General Multiplication Rule

The joint probability that both of two events A and B happen together can be found by

P(A and B) = P(A) P(B|A)

P(B|A) is the conditional probability that B occurs given the information that A occurs.

Definition of Conditional Probability

When P(A)>0, the conditional probability of B given A is

OR P(AB) = P(AB) P(B)

P(B|A) = P(A and B)

P(A)

Multiplication Rule

If one event does not affect the probability of another event, the probability that both events occurs is the product of their individual probabilities.Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent,

P(A and B) = P(A)P(B)

Suppose that 60% of all customers of a large insurance agency have automobile policies with the agency, 40% have homeowner’s policies, and 25% have both types of policies. If a customer is randomly selected, what is the probability that he or she has at least one of these two types of policies with the agency? (Hint: Venn diagram)

Question #3

P(A or B) = P(A) + P(B) – P(A and B)

P(auto or home) = .60 + .40 .25 = .75

Question #6

Drawing two aces with replacement.(2 aces)=P

4

52

4

52

.0059

Drawing three face cards with replacement.

(3 face)=P12

52

12

52

12

52

.0123

Multiplication Rule Practice

Draw 5 reds cards without replacement.

(5 red)=P26

52

25

51

.0253

Draw two even numbered cards without replacement.

(2 even)=P20

52

19

51

.1433

24

50

22

48

23

49

Multiplication Rule Practice

Draw three odd numbered red cards with replacement.

38

(3 red, odd)= .003652

P

Back to Flipchart

Question #71 10 5

2 8 2

Nondefective Defective

Company

Company

What is the probability of a GFI switch from a selected spa is from company 1?

15company 1 .6

25P

What is the probability of a GFI switch from a selected spa is defective?

7defective .28

25P

Question #71 10 5

2 8 2

Nondefective Defective

Company

Company

What is the probability of a GFI switch from a selected spa is defective and from company 1?

5company 1 defective .2

25P

What is the probability of a GFI switch from a selected spa is from company 1 given that it is defective? 5

company 1|defective .71437

P

1 10 5

2 8 2

Nondefective Defective

Company

Company

Question #7

5company 1 defective

25P

5company 1|defective

7P

P(A and B) = P(A) P(B|A)

7defective

25P

5 7 5

25 25 7

RememberTwo events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent,

P(A and B) = P(A)P(B)The joint probability that both of two events A and B happen together can be found by

P(A and B) = P(A) P(B|A)How can we use the formulas to test for independence?

Independence and Mutually Exclusivity

Independence means knowing something about one tells you nothing about the other.Mutually exclusive events cannot happen at the same time.Are independent events mutually exclusive?

Independent Events

Two events A and B that both have positive probability are independent if P(B|A) = P(B)

Back to flipchart

Is the event that a participant is male and the event that he correctly identified tap water independent?

Yes No Total

Male 21 14 35

Female 39 26 65

Total 60 40 100

13. Jack and Jill have finished conducting taste tests with 100 adultsfrom their neighborhood. They found that 60 of them correctlyidentified the tap water. The data is displayed below.

Yes No Total

Male 21 14 35

Female 39 26 65

Total 60 40 100

In order for a participant being male and the event thathe correctly identified tap water to be independent, weknow that

P(male|yes) = P(male) or

P(yes|male) = P(yes)

Yes No Total

Male 21 14 35

Female 39 26 65

Total 60 40 100

In order for a participant being male and the event thathe correctly identified tap water to be independent, weknow that P(yes|male) = P(yes)

21 60 21 60We know P(yes|male) = and P(yes) = . Since

35 100 35 100we can conclude the participant being male and their ability to

correctly identify tap water are independent.

Yes No Total

Male 21 14 35

Female 39 26 65

Total 60 40 100

In order for a participant being male and the event thathe correctly identified tap water to be independent, weknow that P(male|yes) = P(male)

21 35 21 35We know P(male|yes) = and P(male) = . Since

60 100 60 100we can conclude the participant being male and their ability to

correctly identify tap water are independent.

Question #16Has TB Does Not

DNA

DNA

14 0

12 181

14193

26 181 207What is the probability of an individual having tuberculosis given the DNA test is negative? 12

1930622.

P(TB|DNA ) =P(DNA TB)

P(DNA )

12

207193

207.0622

Conditional Probability with Tree Diagrams

17. Dr. Carey has two bottles of sample pills on his desk for the treatment of arthritic pain. He often grabs a bottle without looking and takes the medicine. Since the first bottle is closer to him, the chances of grabbing it are 0.60. He knows the medicine from this bottle relieves the pain 70% of the time while the medicine in the second bottle relieves the pain 90% of the time. What is the probability that Dr. Carey grabbed the first bottle given his pain was not relieved?

stP(1 bottle|pain not relieved)stP(1 bottle not relieved)

P(pain not relieved)

1st

2nd

.6

.4relieved

not

relieved

not

.7

.3

.9

.1

.6 .3 .6 .3 .4 .1 .8182