Theorems of Probability

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UNIT 2 THEOREMS
Structure
2.1 Introduction
Objectives

PROBABILITY

2.2 Some Elementary Theorems
2.3 General Addition Rule

2.4 Conditional Probability and Independence
2.4.1 Conditional Probability 2.4.2 Independent Events and MultiplicationRule 2.4.3 Theorem of Total Probability and Bayes Theorem

2.5 Summary

2.1 INTRODUCTION
You have already learnt about probability axioms and ways to evaluate probability of events in some simple cases. In this unit, we discuss ways to evaluate the probability of combination of events. For this, we derive the addition rule which deals with the probability of union of two events and the multiplication rule which deals with thc probability of intersection of two events. Two important concepts namely : Conditional Probability and independence of events, are introduced and Bayes theorem, which deals with conditional probability is presented.

Objectives
After reading this unit, you should be able to

* * * *

evaluate the probability of certain combination of events involving union, intersection and complementation, evaluate conditional probability, check independence of two or more events, and apply Bayes theorem to find the probability that the "effectY'Awas "caused" by the event B.

2.2 SOME ELEMENTARY THEOREMS
Recall the axiomatic definition of probability which you have read in Section 1.4. Using these axioms of probability, it is possible to derive many results which are very useful in applications. We present some of these results in this section.

Theorem 1 : If 4 is the empty set then

P(4) 0
b=

Proof :
For any eventA, A a A U 9. Also A and 9 are mutually exclusive as A f l 4 = 4. Hence by Axiom 3 of probability axioms,

Probability Concepts

which implies that P ( $ )

-

0.
<

Theorem 2 :

I ~ is the complementary event of A, then A P(A) I-P(A)

-

Proof:

Note that S A U where S is the sample space. FurtherA a n d 2 are $. Therefore by Axiom 3 of probability axioms mutually exclusive asA n P(S) But by Axiom 2, P ( S )

-

A

-- P ( A U A )

-

P(A)+P(A)
i.e.

-

P(A)+P(A) 1 and hence equation ( 2.1 ) gives 1

(2.1)

P ( A )= I - P ( A )

I.L-rem 3 :
If A and B are any two events then P(AUB) P(A)+P(B)-p(AnBf

-

Proof : The below given Venn diagram (Figure 2.1) suggests that eventsA U B and B can be expressed as

AnB

~

n

x

B ( A ~ B ) u ( B ~ A ) AUB - A U ( B ~ A ) But ( A n B ) and ( B n ) are mutually exclusive and hence by Axiom equation (2.2) gives

-

A

P(B)

-

P ( ( A ~ B ) u ( B ~ ~ )( ) A ~ B ) + P ( B ~ ~ ) P

-

Similarly equation (2.3)gives

P ( A U B ) P ( A U ( B ~ ~ ) )( A ) + P ( B ~ A ) P From equation (2.4) and equation (2.5), P(AUB)

-

-

-

P(A)+P(B)-P(AnB)

Remark 1 : TheformulaP(AUB) P ( A ) + P ( B ) - P ( A n B ) i s c a l l e d t h e a d d i t w n rule and is used to evaluate the probability of occurrence of at least one of the two given events. You will learn the general addition rule in Section 2.3.

-

Theorems of Probability

Theorem 4 : IfA andBaretwoeventssuchthatACB, then P ( A ) s P ( B ) ;

Proof:
As A C B, we can express B as the union of two mutually exclusive events

A and ( B fi A ) ( Ref. Figure 2.2) i.e.

Figme 2 2

B = A U ( B ~ ~ ) Axiom 3, together with equation ( 2.6 ) gives

P(B)

-

P ( ' A ) + P ( B ~ ~ )

(2.7)

But P ( B nA ) r 0 (by Axioml) and hence equation (2.7) implies P ( B ) r P(A). ~ o t e t h a t e ~ u a t i o n ( 2 . 7 ) a l s o i m ~ l i e s ~ ((~ ~)~ ) ( A ) , if A C B . P B -P Sometimes we denote the event B fl A by the set theoretic difference B - A and I

-

i

,

so we can write equation (2.7) as P(B-A) provided A C B.

=

P(B)-P(A)

i1
i
1
I

Example 1 :
Suppose that A and B are events for which
P ( A ) = x,P(B)
=

y and P ( A n B ) = z. Express

P ( ~ u B and )

From De-Morgan's laws it is known that ( AU B ) = ( A n B ) and therefore

~...
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