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
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Find the probability that a person stopping at the gas station will have (i) his tyres or his oil checked and (ii) will have neither his tyres nor his oil checked.
2 3 GENERAL ADDITION RULE .
You have already learnt the addition rule ( Theorem 3 ) for finding the probability of occurrence of at least one of the two given events. In this section, you will learn the general addition rule which will help you to evaluate the probability of occurrence of at least one of the given n events. If the a events are denoted by A1 ,A2 ,..An then our aim is to find a formula for evaluating P (A1 U A2 U ... U An ) . For this, let us first take 3 eventsA ,B and C and use Theorem 3 to find the expression for P (A U B U C ) .
Theorem 4 : If A ,B and C are any three events, then
P(AUBUC) = P(A)+P(B)+P(C)-P(AnB)
Tbcorem~ Probability of
i
L
1
Proof : Let us denote B U C by D and therefore
t
where equation (2.9) follows because of Theorem 3. Also using Theorem 3 again,
P(D)
=
P(BUC)
-
P(B)+P(C)-P(BnC)
and
Substituting for P ( D ) and P ( A
n D ) in equation (2.9) we obtain
which proves
Option | Probability of Occurring | 2011 | 2012 | 2013 | 2014 | 2015 | Total |
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