If `A` and `B` are two independent events, prove that `P(AuuB).P(A'nnB')<=P(C)`, where `C` is an event defined that exactly one of `A` and `B` occurs.

Answer

A

Explanation

Here, `P(AuuB).P(A'nnB')`
`implies{P(A)+P(B)-P(AnnB)}{P(A').P(B')}`
`["since A, B are independent, so A',B' are independent"]`
`:.P(AuuB).P(A'nnB')le{P(A)+P(B)}.{P(A').P(B')}`
`=P(A).P(A').P(B')+P(B).P(A').P(B')`
`leP(A).P(B')+P(B).P(A')" "...(i)`
`[becauseP(A')le1" and "P(B')le1]`
`impliesP(AuuB).P(A'nnB')leP(A).P(B')+P(B).P(A')`
`impliesP(AuuB).P(A'nnB')leP(C)`
`[becauseP(C)=P(A).P(B')+P(B).P(A')]`