Let be a sample space in which two events and satisfy and

MATHEMATICS · Multi-part · A Level · Mathematics · Probability · Probability

Problem

Let

S

be a sample space in which two events

A

and

B

satisfy

P(A)=0.5

and

P(B)=p

, where

0<p<1

. Suppose that

P(A\cup B)=0.7

. (i) Express

P(A\cap B)

in terms of

p

. (ii) Using the definition of independence, obtain the condition on

p

such that

A

and

B

are independent. (iii) For the value of

p

found in part (ii), prove that the complements

A'

and

B'

are also independent. (iv) For the same value of

p

, verify whether

A

and

B'

are independent. Show all steps clearly.