Mathematics

Problem

For two events A and B in a sample space,

P(A)=0.55

and

P(B)=0.45

. a. (i) Using only the axioms of probability, determine the greatest and the least possible values of

P(A\cap B)

. b. (ii) Find the value of

P(A\cap B)

that would make A and B independent. c. (iii) For the value obtained in part (ii), compute

P(A\mid B)

and

P(B\mid A)

and interpret each in one sentence. d. (iv) If instead

P(A\cap B)=0.25

, decide whether A and B are independent and justify your answer with calculation.