If M and N are any two events, the probability that atleast one of them occurs is

Options

A. `P(A)+P(B)-2p(A cap B)`
B. `P(A cap overline(B))+P(overline(A) cap B)`
C. `P(A cap B)-P(A cap B)`
D. `P(overline(A))+P(overline(B))-2P(overline(A) cap overline(B))`

Answer

D

Explanation

We have,
P(Exactly one of A, B occurs)
`=P[(A cap overline(B)) cup (overline(A) cap B)]`
`=P(A cap overline(B))+P(overline(A) cap B)`
`=P(A)-P(A cap B)+P(B)-P(A cap B)`
`=P(A)+P(B)-2P(A cap B)=P(A cap B)-P(A cap B)`
Also,
P, (Exactly one of A, B occurs)
`=P(A cup B)-P(A cap B)`
`={1-P(overline(A cup B))}-{1-P(overline(A cap B))}`
`=[1-P(overline(A) cap overline(B))]-[1-P(overline(A) cup overline(B))]`
`=P(overline(A) cup overline(B))-P(overline(A) cap overline(B))=P(overline(A))+P(overline(B))-2P(overline(A) cap overline(B))`.
Hence, option (d) is incorrect.