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Let H1,H2,………Hn be mutually exculusive e...

Let `H_1,H_2,………H_n` be mutually exculusive events with `P(H_i)gt 0,i=1,2`,……,n. Let E be any other event with `0lt P€ lt1`.
Statement -1 `P(H_i//E)gt P(E//H_i)P(H_i)`, for `i=1,2…..,n`.
Statement -2 ` sum_(i=1)^(n) P(H_i)=1`

A

Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement-2 is True.

Text Solution

Verified by Experts

The correct Answer is:
D
Since `H_(1), H_(2),H_(3) ,…., H_(n) ` are exhaustive events.
`overset(n)underset(i=1)cupH_(i) ` = S , where S is the sample space
` P (overset(n)underset(i=1)cup H_(i)) = P(S)`
`Rightarrow overset(n)underset(i=1)sum P (H_(i))=1 `
So, satatement -2 is true.
Now, two cases arise.
CASE I when ` H_(i) cap E = varphi " for some " H(i)`
In this case, we have
` Rightarrow P (E//H_(i)) P ( H_(i) ) =0`
and ` P ( H_(i) // E) = ( P (H _(i) cap E))/( P(E)) ` = 0 for some i
So, statement -1 is not true for such values of i
CASE II when ` 0 gt P ( H_(i) cap E) gt 1 " for " i=1,2 ,..., n` ,
In this case, we have
` P ( H_(i) //E) = (P (H_(i) cap E))/ (P(E)) = ( P (H_(i) P ( E// H_(i)) /( P (E))`
` Rightarrow P ( H_(i) //E) gt P(H_(i) P ( E //H_(i))`
Hence, statement -1 is not always true.
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