Mathematics

Problem

Let

A

and

B

be two events such that

P(A)=0.55

and

P(A\cup B)=0.80

. a. (i) Given that

P(A\cap B)=0.15

, find

P(B)

. b. (ii) Suppose

P(A\cap B)=x

. Using only the axioms of probability and the relationship among

P(A)

P(B)

P(A\cup B)

and

x

, determine the range of possible values of

x

. c. (iii) For the maximum possible value of

x

obtained in part (ii), calculate the corresponding value of

P(B)

and comment on the nature of the relationship between

A

and

B

in that case.