1.

In Example 94, if n is even and E denotes the event of choosing even numbered urn `(p(U_(i))=(1)/(n))`, then the value of `P(W//E)`, isA. `(n+2)/(2n+1)`B. `(n+2)/(2(n+1))`C. `(n)/(n+1)`D. `(1)/(n+1)`

Answer» Correct Answer - B
We have, `P(U_(i))=(1)/(n) " for " i=1,2,3,..,n`
Clearly,
`P(W//E)=(P(W cap E))/(P(E ))`
Now,
`P(W cap E)=P(W cap U_(2))+P(W cap U_(4))+P(W cap U_(6))+..+P(W cap U_(n))`
`implies P(W cap E)=P(U_(2))P(W//U_(2))+P(U_(4))P(W//U_(4))+..+ (1)/(n)xx(n)/(n+1)`
`implies P(W cap E)=(1)/(n)xx(2)/(n+1)+(1)/(n)xx(4)/(n+1)+(1)/(n)xx(6)/(n+1)+..+ P(U_(n))P(W//U_(n))`
`implies P(W cap E)=(2+4+6+..+n)/(n(n+1))=(n(n+2))/(4n(n+2))=(n+2)/(4(n+1))`
and `P(E )=P(U_(2))+P(U_(4))+..+4(U_(n))`
`implies P(E )=(1)/(n)+(1)/(n)+..+(1)/(n)((n)/(2) " times")`
`implies P(E )=(1)/(2)`
`therefore P(W//E)=(n+2)/(2(n+1))`
Total Number of white balls in even
ALITER `P(W//E)=("numbered urns")/("Total number of balls in even numbered urns")`
`implies P(W//E)=(2+4+6+..+n)/((n)/(2)(n+1))=((n)/(4)(n+2))/((n)/(2)(n+1))=(n+2)/(2(n+1))`


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