An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (A) 2, 4 or 8 (B) 3, 6 or 9

Options

A. 2,4 or 8
B. 3,6 or 9
C. 4 or 8
D. 5 or 10

Answer

D

Explanation

Since , `P(A)=(2)/(5)`
For independent events , `P(A cap B )= P(A)P(B)`
`implies P(A cap B ) le (2)/(5)`
`implies P(A cap B ) =(1)/(10),( 2)/(10), (3)/(10),(4)/(10) ` [maximum 4 outcomes may be in `A cap B` ]
(i) Now , `P(A cap B )=(1)/(10) `
`implies P(A) *P(B) =(1)/(10) `
`implies P(B)=(1)/(10) xx (5)/(2)=(1)/(4)` , not possible
(ii) Now , `P(A cap B ) =(2)/(10) implies (2)/(5) xx P(B) =(2)/(10)`
`implies P(B) =(5)/(10) `, outcomes of B=5
(iii) , `P(A cap B ) =(3)/(10)`
`implies P(A)P(B) =(3)/(10) implies (2)/(5) xx P(B) =(3)/(10) `
`P(B) =(3)/(4)` , not possible
(iv) Now , `P(A cap B ) =(4)/(10) implies P(A) *P(B) =(4)/(10)`
`implies P(B) =1 `, outcomes of B=10