Events A and B are such that State whether A and B are independent? - Wired Faculty

Question

Events A and B are such that $straight P left parenthesis straight A right parenthesis space equals space 1 half comma space straight P left parenthesis straight B right parenthesis space equals space 7 over 12 space and space straight P left parenthesis not space straight A space or space not space straight B right parenthesis space equals 1 fourth.$ State whether A and B are independent?

Solution

straight P left parenthesis straight A right parenthesis space equals space 1 half comma space space straight P left parenthesis straight B right parenthesis space equals space 7 over 12

Now P(not A or not B) =

1 fourth space space space rightwards double arrow space space space space straight P left parenthesis straight A to the power of straight c space or space space straight B to the power of straight c right parenthesis space equals space 1 fourth

rightwards double arrow space space space space straight P left parenthesis straight A to the power of straight c space union space straight B to the power of straight c right parenthesis space equals space 1 fourth space space space rightwards double arrow space space space space straight P open square brackets left parenthesis straight A space intersection space straight B right parenthesis to the power of straight c close square brackets space equals space 1 fourth rightwards double arrow space space space space 1 minus space straight P left parenthesis straight A space intersection space straight B right parenthesis space equals space 1 fourth space space space space rightwards double arrow space space space straight P left parenthesis straight A space intersection straight B right parenthesis space equals space 1 minus 1 fourth space equals space 3 over 4

Also,

straight P left parenthesis straight A right parenthesis space cross times space straight P left parenthesis straight B right parenthesis space equals space 1 half cross times 7 over 12 space equals space 7 over 24 not equal to 3 over 4

space space space space straight P left parenthesis straight A right parenthesis space cross times space straight P left parenthesis straight B right parenthesis space equals space 1 half cross times 7 over 12 space equals 7 over 24 not equal to 3 over 4

therefore space space space space straight P left parenthesis straight A right parenthesis space cross times space straight P left parenthesis straight B right parenthesis space space not equal to space straight P left parenthesis straight A intersection straight B right parenthesis

∴ A and B are not independent.

For Daily Free Study Material Join wiredfaculty