Question

There are $n$ urns each containing $n+1$ balls such that the ith um contains $i$ white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and $w$ denotes the event of getting a white ball.

If $n$ is even and $E$ denotes the event of choosing even numbered um $(P\left(u_i\right)=\frac{1}{n})$, then the value of $P\left(w/E\right)$ is

• A. $\frac{n+2}{2n+1}$
• B. $\frac{n+2}{2(n+1)}$
• C. $\frac{n}{n+1}$
• D. $\frac{1}{n+1}$

Answer 1

The correct option is B $\frac{n+2}{2(n+1)}$

given that $n$ is even
$n(w\cap E)=$ event of getting white balls from even numbered urn
$=2+4+6+......+n=2(1+2+3+....+\frac{n}{2})=2\times \frac{\left(n/2\right)(n/2+1)}{2}=\frac{n}{4}(n+2)$
$n\left(E\right)=$ event of selecting even numbered urn
$=$ no. of balls in each urn $\times$ number of even numbered urns
$=(n+1)\times \frac{n}{2}$
$P\left(\frac{w}{E}\right)=\frac{n(n+2)/4}{n(n+1)/2}=\frac{n+2}{2(n+1)}$