1.

If A and B are two events such that `P(A)=(1)/(2)` and `P(B)=(2)/(3)`, then which of the following is incorrect ?A. `P(A cap B) ge (2)/(3)`B. `P(A cap overline(B))+ ge (1)/(3)`C. `(1)/(6) le P(A cap B) le (1)/(2)`D. `(1)/(6) le P(overline(A) cap B) le (1)/(2)`

Answer» Correct Answer - B
We have,
`P(A cap B) ge " max" {P(A),P(B)}=2//3`
and, `P(A cap B) le " min" {P(A),P(B)}=1//2`
So, option (a) is correct.
Now,
`P(A cap B)=P(A)+P(B)-P(A cup B)`
`implies P(A cap B) ge P(A)+P(B)-1=(1)/(6)`
`therefore (1)/(6) le P(A cap B) le (1)/(2)`
So, option (c ) is correct.
Now, `(1)/(6) le P(A cap B) le (1)/(2)`
`implies -(1)/(2) le -(P cap B) le -(1)/(6)`
`implies (2)/(3)-(1)/(2) le P(B)-P(A cap B) le(2)/(3)-(1)/(6) le P(overline(A) cap B) le (1)/(2)`
So, option (d) is correct.
Now, `P(A cap overline(B))=P(A)-P(A cap B)`
`implies P(A cap overline(B)) le (1)/(2)-(1)/(6)=(1)/(3)`
So, option (d) is correct.


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