For the following questions, choose the correct answer from the codes (a),(b),(c ) and (d) defined as follows.
Let `H_(1),H_(2),.........H_(n)` be mutually exclusive events with `P(H_(i)) gt 0`,i=1,2,.......n,.Let E be any other event with `0 lt P(E ) lt 1`.
Statement I `P(H_(i)|E ) gt P(E|H_(i)).P(H_(i))` for
i=1,2,......,n
Statement II `sum_(i=1)^(n) P(H_(i))=1`

Options

A. Statement I is true, Statement II is also true, Statement II is the the correct explanation of Statement I
B. Statement I is true, Statement II is also true, Statement II is is not the correct explanation of Statement I
C. Statement I is true,Statement II is false
D. Statement I is false, Statement II is true

Answer

D

Explanation

Statement I if `P(H_(i)nnE)=0` for some I, then
`P((H_(i))/(E))=P((E)/(H_(i)))=0`
If `P(H_(i)nnE)neO,AA " " i=1,2,...,n,`then
`P((H_(i))/(E))=(P(H_(i)nnE))/(P(H_(i)))xx(P(Hi))/(P(E))`
`=(P((E)/(H_(i)))xxP(H_(i)))/(P(E))gtP((E)/(H_(i)))*P(H_(i))" " [therefore OltP(E)lt1]`
Hence, Statement I may not always be true.
Statement II Clearly, `H_(1)uuH_(2)uu...uuH_(n)=S`
[same space ]
`rArr " " P(H_(1))+P(H_(2))+...+P(H_(n))=1`
Hence, Statement II is true.