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Sachin Kaushal, Lovely Professional University Unit 3: Conditional Probability and Independence Baye’s Theorem

Unit 3: Conditional Probability and Notes

Independence Baye’s Theorem

CONTENTS
Objectives
Introduction

3.1 Conditional Probability
3.2 Baye’s Theorem
3.3 Independence of Events
3.4 Repeated Experiments and Trials
3.5 Summary

3.6 Keywords
3.7 Self Assesment
3.8 Review Questions

3.9 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss probabilities to the outcomes of a random experiment with discrete sample space,

 Explain properties of probabilities of events, I
 Describe the probability of an event,
 Explain conditional probabilities and establish Bayes theorem,

Introduction

Suppose that two series of tickets are issued for a lottery. Let 1,2,3,4,5 be the numbers on the 5
tickets in series I and let 6,7, 8,9, be the numbers on the. 4 tickets in series 11. I hold the ticket,
bearing number 3. Suppose the first prize in the lottery is decided by selecting one of the
5 + 4 = 9 numbers at random. The probability that I will win the prize is 1/9. Does this probability
change if it is known that the prize-winning ticket is from series I? Ineffect, we want to know the
probability of my winning the prize, conditional on the knqwledge that the prize-winning
ticket is from series I.

In order to answer this question, observe that the given information reduces our sample-space
from the set { 1,2,3,.4,5,6,7,8,9 } to its subset { 1.2.3.4.5 } containing 5 points. In fact, this subset
{ 1,2,3,4,5 } corresponds to the event H that the prize winning ticket. belongs to series I. If the
prize winning ticket is selected by choosing one of these 5 numbers at random, the probability
that I will win the prize is 115. Therefore, it seems logical to say that the conditional probability
of the event A of my winning the prize, given that the prize-winning number is from series I, is
P(A | H) = 1/5.

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