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Chapter 11 ProbabilityWeek 12 & 13

Lecture 5: Multiplicative Law Total Probability Theorem Bayes’ Theorem

Apr 10, 2023

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Learning Objectives

Apply multiplicative law to fine probability of certain events.

Apply a total probability rule to find the probability of an event when the event is partitioned into several mutually exclusive and exhaustive subsets.

Apply Bayes’ theorem to find the conditional probability of an event when the event is partitioned into several mutually exclusive and exhaustive subsets.

Apr 10, 2023

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Multiplicative Law of Probability and Independence

( ) ( ). ( )P A B P A P B

( ) ( | ). ( )P A B P A B P B For two events A and B

Definition: Events A and B are independent if and only if

If events A1, .., Ak are independent then,

1 2 1 2( ... ) ( ) ( ) ( )k kP A A A P A P A P A

Apr 10, 2023

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Multiplication rule is most useful when the experiment consists of several stages in succession. The conditioning event, B, describes the outcome of the first stage and A is the outcome of the second, so that P( A| B) – conditioning on what occurs first – will often be known.

Apr 10, 2023

EXAMPLE 14.

Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects (88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%). Electrical connect defects are caused by defective wires (35%), improper connections (13%) or poorly welded wires (52%).

Find the probability

a) that a failure is due to loose keys.

b) that a failure is due to improperly connected or poorly welded wires.

5Apr 10, 2023

Jan 2009 6

Example 14Computer keyboard failures are due to faulty electricalConnects (12%) or mechanical defects (88%). Mechanicaldefects are related to loose keys (27%) or improperassembly (73%). Electrical connect defects are caused bydefective wires (35%), improper connections (13%) or poorlywelded wires (52%).

Find the probability • that a failure is due to loose keys.

• That a failure is due to improperly connected or poorly welded wires.

(0.88)(0.27) = 0.237

(0.12)(0.13+0.52) = 0.0078

EXAMPLE 15.

During a space shot, the primary computer system is backed up by two secondary systems. They operate independently of one another, and each is 90% reliable. What is the probability that all three systems will be operable at the time of the launch?

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Jan 2009 8

Example 15 During a space shot, the primary computer system is backed up by two secondary systems. They operate independently of one another, and each is 90% reliable. What is the probability that all three systems will be

operable at the time of the launch?

Solution

A1: event main system is operableA2: event first backup is operableA3: event second backup is operable

Given P(A1) = P(A2) = P(A3) = 0.9Since they operate independentlyP(A1 ∩ A2 ∩A3) = P(A1)P(A2) P(A3) = 0.729

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The Law of Total Probability

Suppose B1, B2 ,…, Bn are mutually exclusive and exhaustive in S, then for any event A

1 1

( ) ( ) ( | ) ( )n n

i i ii i

P A P A B P A B P B

B1

SA

A∩ B1

A∩ B3A∩ B4A∩ B2

B1 B2B3

B4

Apr 10, 2023

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Bayes’ Theorem

Suppose B1, B2,…, Bn are mutually exclusive and exhaustive (whose union is S). Let A be an event such that P(A) > 0. Then for any event Bj , j =1, 2, …, n,

1

( ) ( | ) ( )( | )

( ) ( | ) ( )

k k kk n

i ii

P A B P A B P BP B A

P A P A B P B

Apr 10, 2023

EXAMPLE 16.

A particular city has three airports. Airport A handles

50% of all airline traffic, while airports B and C handle

30% and 20%, respectively. The rates of losing a

baggage in airport A, B and C are 0.3, 0.15 and 0.14

respectively. If a passenger arrives in the city and losses

a baggage, what is the probability that the passenger

arrives at airport A?

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Jan 2009 12

Example 16. A particular city has three airports. Airport A handles 50% of all airline traffic, while airports B and C handle 30% and 20%, respectively. The rates of losing a baggage in airport A, B and C are 0.3, 0.15 and 0.14 respectively. If a passenger arrives in the city and losses a baggage, what is the probability that the passenger arrives at airport A?

Let L = event of losing a baggage and A = event arriving at Airport A

)|()()|()()|()(

)|()()|(

CLPCPBLPBPALPAP

ALPAPLAP

)14.0)(2.0()15.0)(3.0()3.0)(5.0(

)3.0)(5.0(

673.0

Solution

EXAMPLE 17.

A company rated 75% of its employees as

satisfactory and 25% unsatisfactory. Of the

satisfactory ones 80% had experience, of the

unsatisfactory only 40%. If a person with

experience is hired, what is the probability that

(s)he will be satisfactory?

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Jan 2009 14

Example 17 A company rated 75% of its employees as satisfactory

and 25% unsatisfactory. Of the satisfactory ones 80% had experience, of the unsatisfactory only 40%. If a person with experience is hired, what is the probability that (s)he will be satisfactory?

Let E = experience, N = no experience, S = Satisfactory, U = unsatisfactory

P( E | S ) = 0.8; P( E | U ) = 0.4, P ( N | S) = 0.2; P( N | U ) = 0.6, P(S) = 0.75, P(U) = 0.25

Q: what is P( S | E) ?

Ans: 0.857

Solution

Jan 2009 15

Example 18: In a certain assembly plant, three machines, B1, B2, B3, are produced 30%, 45% and 25%, respectively, of the products. It is known from past experience that 2%,3% and 2% of the products made by each machine, respectively, are defective. Now, suppose that a finished product is randomly selected. What is the probability that it is defective?Solution:

0245.0005.00135.0006.0

)02.0)(25.0()03.0)(45.0()02.0)(3.0(

)B|D()B()B|D()B()B|D()B(D)( 332211

PPPPPPP

If a product was chosen randomly and found to be defective, what is the probability that it was made by machine B3?

49/100245.0/005.00245.0/)]02.0)(25.0[(

)(/)]|()([)|B( 333

DPBDPBPDP

QUIZ 7

From a total of 50 groups of ETP, 15 groups are not qualified

for the EDX2011. Among the remaining 35 groups, each group has

either potential to win a gold price or a special award. Let say that 15

groups have a chance to win a gold price and 25 groups have a chance

to win a special award. Let A be the event of wining a gold price and B

be the event of wining the special award. If one group is selected

randomly, what is the probability that the group has wining

(i) both, the gold and the special award?

(ii) the gold price only?

(iii) at least one of the price?

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