Question

P(A) = 12, P(B) = 13 and P(A ∩ B) = 16. Are the events A and B independent?

Answer 1

Answer:

Let A', B' denote the complements of A, B respectively.

Given that, P(A∩B')=1/6

or, P(A)P(B')=1/6

or, P(A)[1-P(B)]=1/6

or, P(A)=1/6[1-P(B)]  ------------------------(1) and

P(A'∩B)=2/15

or, P(A')P(B)=2/15

or, [1-P(A)]P(B)=2/15

or, [1-1/6{1-P(B)}]P(B)=2/15

or, [{6-6P(B)-1}/{6-6P(B)}]P(B)=2/15

or, 15[5-6P(B)]P(B)=2[6-6P(B)]

or, 15[5-6P(B)]P(B)=12[1-P(B)]

or, 5[5-6P(B)]P(B)=4[1-P(B)]

or, 25P(B)-30[P(B)]²=4-4P(B)

or, -30[P(B)]²+25P(B)+4P(B)-4=0

or, 30[P(B)]²-29P(B)+4=0

or, 30a²-29a+4=0 where P(B)=a (say)

or, 30a²-24a-5a+4=0

or, 6a(5a-4)-1(5a-4)=0

or, (6a-1)(5a-4)=0

∴, either, 6a-1=0

or, 6a=1

or, a=1/6

or, P(B)=1/6

Or, 5a-4=0

or, 5a=4

or, a=4/5

or, P(B)=4/5

∴, P(B)=1/6, 4/5 Ans.

follow me

thank me