If `A and B` are two events, the probability that exactly one of them occurs is given by
a. `P(A)+P(B)-2P(AnnB)` b. `P(Ann B )+P( AnnB')` c. `P(AuuB)-P(AnnB)` d. `P( A' )+P( B' )-2P( A' nn B' )`

Options

A. `P(A) + P(B) - 2P(A nn B)`
B. `P(A nn barB) + P(barA nn B)`
C. `P(A uu B) - P(A nn B)`
D. `P(barA) + P(barB) - 2P(barA nn barB)`

Answer

A, B, C, D

Explanation

We have,
P(exactly one of A, B occurs)
`= P [(A nn barB) uu (barA nn B)]`
`=P(A nn barB) + P(barA nn B)`
`=P(A) - P(A nn B) + P(B) - P(A nn B)`
`=P(A) + P(B) - 2P(A nn B)`
`=P(A uu B) - P(A nn B)`
Also,
P(exactly one of A, B occurs)
`=[1 - P(barA nn barB)] - [1 - P(barA uu barB)]`
`=P(barAuubarB) - P(barA nn barB) = P(barA) + P(barB) - 2P(barA nn barB)`