Probability Theory: Conditional Probability, Derangements, and Venn Diagrams, Slides of Engineering Mathematics

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Conditional probability

Midterm

• 22 nd^ Mar

• One and a half hour.

• Bring calculator and blank papers.

• Close-book and close-note exam.

• Coverage:

– Lecture 1 to 15.

– Tutorial 1 to 7.

– Homework 1 to 3.

Venn diagram 

A 1 A 2

A 3

A 4

The case n=4 (cont’d)

• A 1  A 2 = {1234, 1243}.

• A 1  A 3 = {1234, 1432}.

• A 1  A 4 = {1234, 1324}.

• A 2  A 3 = {1234, 4231}.

• A 2  A 4 = {1234, 3214}.

• A 3  A 4 = {1234, 2134}.

• A 1  A 2  A 3 = A 1  A 2  A 4 = A 1  A 3  A 4 =A 2  A 3  A 4

= A 1  A 2  A 3  A 4 ={1,2,3,4}.

• By PIE, the number of derangements for n=4 is equal to

• Indeed, the nine derangements for n=4 are

John Venn (1834-1923)

• British logician and philosopher

• http://en.wikipedia.org/wiki/John_Venn

Classical definition of probability

• From Laplace’s Théorie

analytique des probabilités

– “The probability of an event is

the ratio of the number of

cases favourable to it, to the

number of all possible cases.”

http://en.wikipedia.org/wiki/Pierre-Simon_Laplace

Disjoint events

• Two events are called disjoint if the

intersection is empty, i.e., if they have no

overlap.

• The calculation of union of events in general is

complicated when the events overlap.

• It is much easier if we want to compute the

probability of a union of disjoint event. We

simply add the probability of the events

Union of disjoint events

A

B

C

A, B, C mutually disjoint Pr(A  B  C) = Pr(A) + Pr(B) + Pr(C).

Law of total probability



A

B

C

D

Pr(E) = Pr(EA)+ Pr(EB)+ Pr(EC)+ Pr(ED).

A  B  C  D = ,

A, B, C, D mutually disjoint

E

CONDITIONAL PROBABILITY

Conditional probability

• If we are given an addition information that the outcome is in event B,

then we can update the likelihood to |AB|/ |B|.

• If Pr(B) is nonzero, then we define the conditional probability of A given B

by

Pr(A|B) = Pr(AB)/Pr(B).

A

B

At the beginning, the probability of event A is |A| / ||, assuming that all outcomes in  are equally likely.

The law of total probability in

terms of conditional probability



A

B

C

D

Pr(E) = Pr(EA)+ Pr(EB)+ Pr(EC)+ Pr(ED)

= Pr(E|A)Pr(A)+ Pr(E|B)Pr(B)+ Pr(E|C)Pr(C)+ Pr(E|D)Pr(D).

A  B  C  D = ,

A, B, C, D mutually disjoint

E

Independent events

• If P(A|B) = P(A), then it means that the likelihood

of the event A does not change after we are told

that event B has occurred. In this case, event A is

said to be independent of event B.

• As Pr(A|B) = Pr(AB)/Pr(B), event A is

independent of B if Pr(AB) = Pr(A) Pr(B).

• Two events A and B are said to be independent,

or statistically independent, if

Pr(AB) = Pr(A) Pr(B).

Example

• Roll two fair dice.

• Let A be the event that the face value of the first die is

even.

• Let B be the event that the sum of the dice is odd.

• Are events A and B independent?

First number is even

Sum is odd