• The probability of an event B occurring when it is known that some event A has occurred is called a conditional probability.

• Definition:

• The conditional probability of B, given A and denoted by is defined by provided

Conditional Probability

AP

BAPABP

/ ABP /

0AP

Consider an event B of getting a perfect square when a die is tossed.The die is constructed so that even numbers are twice as likely to occur as odd numbers.That is P(odd)=p , P(even)=2pso p=1/9Now suppose that the toss of the die resulted in a number greater that 3, which reduced our sample space S={4,5,6}P(B occurs relative to reduced sample space A) isP(4)+P(5)+P(6)=12w+w+2w=1w=1/5now P(odd)=1/5 and P(even)=2/5P(B/A i.e number is a perfect square given that it is greater that 3)=2/5

ABP /

• Suppose sample space S is the population of adults in a small town who have completed the requirements for a college degree.

The data is categorized according to gender and employment status. One of these individuals is selected at random for a tour throughout the country to publicize the advantages of establishing new industry in the town.

Employed Unemployed Total

Male 460 40 500

Female 140 260 400

Total 600 300 900

• M: a man is chosen• E: The one chosen is employed.

Find P(M/E), probability that a man is chosen given that he is employed.

as

SnEn

SnMEn

En

EMnEMP

/

//

30

23

32

4523

/

EP

EMPEMP

3

2

900

60045

23

900

460

EP

MEP

The probability that a

regularly scheduled flight departs on time is P(D)=0.83;

The probability that it arrives on time is P(A)=0.82;

The probability that it departs and arrives on time is

Find the probability that a plane

a) Arrives on time given that it departed on time.

b) Departed on time given that it has arrived on time.

78.0 ADP

DP

ADPDAP

/

94.083.0

78.0

95.0

82.0

78.0/

AP

ADPADP

Consider an industrial process in the textile industry in which strips of a particular type of cloth are being produced. These strips can be defective in two ways, length and nature of texture. It is known from the historical information on the process that 10% of strips fail the length test, 5% fail the texture text, and only 0.8% fail both tests. If a strip is selected randomly from the process and a quick measurement identifies it as failing the length test, what is the probability that it is texture defective?

Sol: Let L: length defective, T: Texture defective.

Given that the strip is length defective, the probability that the strip is texture defective is given by

08.0

1.0

008.0/

LP

LTPLTP

Independent Events: Two events A and B are independent if and only if

or

assuming the existences of conditional probability.. Otherwise, A and B are dependent.

Multiplicative RulesTheorem: If in an experiment the events A and B can both occur, then

provided P(A)>0.

Since the events and are equivalent, we can also write

,/ BPABP ./ APBAP

./ ABPAPBAP

./ BAPBPABP

BA AB

Suppose that we have a fuse box containing 20 fuses, of which 5 are defective. If 2 fuses are selected at random and removed from the box in succession without replacing the first, what is the probability that both fuses are defective?

Sol:

Let A be the event that first fuse is defective.

B the event that second fuse is defective.

Then is that A occurs, and B occurs after A has occurred.

Probability of selecting first defective bulb

BA

(probability of selecting first defective bulb).

(probability of selecting second defective

bulb from remaining four)

⇒

(probability of selecting

both defective fuses).

4

1

20

5AP

19

4/ ABP

19

1

19

4

4

1

BAP

Theorem:two events A and B are independent iff

Therefore to obtain the probability of that two events both occur, we simply find the product of their individuals.

Example:A small town has one fire engine and one ambulance available for engineers. The probability that the fire engine is available when needed is 0.08 and probability that ambulance is available when called is 0.92. In the event of an injury resulting from a burning find the probability that the both ambulance and the fire engine will be available.

Sol:

=(0.08)(0.92)=0.916

)()( BPAPBAP

)()( BPAPBAP

An electrical system consists of four components . The system works if components A and B work and either of the components C or D work. The reliability of each component is given. Find the probability that

a) Entire system works,

b) The component C does not work, given that entire system works.

Assume that four components work independently.

Sol:a) Entire system work if either of C or D works.P[A ∩ B ∩ (CUD)]=P(A)P(B)P(CUD) =P(A)P(B)[1-P(C′ ∩ D′)]Since P(C′)=P(D′)=1-0.8=0.2so probability becomes=(0.9)(0.9)[1-(0.2)(0.2)]=0.7776

b) Component c does not work given entire system works=P(A ∩ B ∩ C′ ∩ D)/P(entire system works)=P(A)P(B)P(C′)P(D)/0.7776=(0.9)(0.9)(0.2)(0.8)/0.7776

Two cards are drawn from a well shuffled ordinary deck of 52 playing cards. Find the probability that they are both aces if the first card is (i) replaced, (ii) not replaced.Sol:Let A be the event Ace on the first draw and B be the event Ace on the second draw.In Case of replacement, event A and B are independent, thusP(Both are Aces)=P(A∩B)=P(A)P(B)=4/52)(4/52)

If the first card is not replaced, then events are dependents soP(Both are Aces)=P(first card is an Ace)P(Second card is an Ace given that the first card is an Ace)soP(A∩B)=P(A)P(B/A)=(4/52)(3/51)

Theorem: If, in an experiment, the events

can occur, then

If events are independent, then

kAAA ,...,, 21

...//... 21312121 AAAPAAPAPAAAP k 121 .../ kk AAAAP

kAAA ,...,, 21

....... 32121 kk APAPAPAPAAAP

Two fair dice one red and one green are thrown. Let A denote the event that the red die shows an even number and B, the event the green die shows a 5 or 6. Show that the events A and B are independent.

Sol:The sample space S contains 36 equally likely outcomes.A= event that red shows an even numberB= event that green shows 5 or 6A∩B= event that as both events P(A)=18/36P(B)=12/36P(A∩B)=6/36Since P(A∩B)=1/6=(1/2)(1/3)=P(A)P(B)Therefore both events are independent

A random sample of 200 adults are classified below by sex and their level of education attained.Education Male FemaleElementary 38 45Secondary 28 50College 22 17If a person is picked at random from this group, find the probability that(a) the person is a male, given that the person has a secondary education;(b) the person does not have a college degree, given that the person is a female.

If R is the event that a convict committed armed robbery and D is the event that the convict pushed dope, state in words what probabilities are expressed by(a) P(R|D);(b) P(D’|R);(c) P(R|’D).

A class in advanced physics is composed of 10 juniors, 30 seniors, and 10 graduate students. The final grades show that 3 of the juniors, 10 of the seniors, and 5 of the graduate students received an A for the course. If a student is chosen at random from this class and is found to have earned an A, what is the probability thathe or she is a senior?

In an experiment to study the relationship of hypertension and smoking habits, the following data are collected for 180 individuals:

Where H and NH stand for hypertension and non hypertension, respectively. If one of these individuals is selected at random, find the probability that the person is

a) experiencing hypertension, given that the person is a heavy smoker.

b) a non smoker, given that the person is experiencing no hypertension.

Nonsmokers Moderate Smokers

Heavy Smokers

Total

H 21 36 30 87

NH 48 26 19 93

Total 69 62 49 180

For married couples living in a certain suburb, the probability that a husband will vote on a bond referendum is 0.21, the probability that his wife will vote in referendum is 0.28, and the probability that both husband and wife will vote is 0.15. Find the probability that

a) At least one member of a married couple will vote?

b) A wife will vote, given that her husband will vote?

c) A husband will vote given that his wife does not vote?

Before the distribution of certain statistical software every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rate for the four testing programs are, respectively, 0.01, 0.03, 0.02 and 0.01.

a) What is the probability that a CD was tested and failed anyway?

b) Given that a CD was tested, what is the probability that it failed program 2 or 3?

c) In a sample of 100, how many CDs would you expect to be rejected?

The probability that a person visiting his dentist will have an X-ray is 0.6; the probability that a person who has an X-ray will also have a cavity filled is 0.3; and the probability that a person who has had an X-ray and a cavity filled will also have a tooth extracted is 0.1.What is the probability that a person visiting his dentist will have an X-ray, a cavity filled, and a tooth extracted?