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If A and B are any two events, then the ...

If A and B are any two events, then the probability that exactly one of them occur is

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)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
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)`
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Knowledge Check

  • If A and B are any two events, the probability that exactly one of them occurs is

    A
    `P(A) + P(B) + 2P(A cap B)`
    B
    `P(A) + P(B)-P(A cap B)`
    C
    `P(barA)+P(barB)+2P(barA cap barB)`
    D
    `P(A cap barB)+P(barA cap barB)`

  • If A and B are any two events then P(barAnnB) =

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

  • If A and B are any two events, then P(A cap B')= 

    A
    P(A) + P(B')
    B
    `P(A)+P(A cap B)`
    C
    `P(B)-P(A cap B)`
    D
    `P(A)- P( A cap B)`

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