Answer the following questions about conditional probability and the multiplication rule.

MATHEMATICS · Multi-part · CBSE · Mathematics · Probability · Probability

Problem

Answer the following questions about conditional probability and the multiplication rule. (i) Define the conditional probability of an event

A

given another event

B

with

P(B)\neq 0

. (ii) Starting from your definition in part (i), derive the multiplication rule that links

P(A\cap B)

with

P(A)

and

P(B \mid A)

. (iii) Explain how the result obtained in part (ii) extends to three events

E_1,E_2,E_3

and write the general algebraic form for

P(E_1\cap E_2\cap E_3)

in terms of successive conditional probabilities. (iv) For events

E_1,E_2,E_3

with

P(E_1)=0.6

P(E_2 \mid E_1)=0.5

and

P(E_3 \mid E_1\cap E_2)=0.4

, write the explicit expression for

P(E_1\cap E_2\cap E_3)

and interpret, in words, the meaning of each factor in the product without carrying out any numerical multiplication.