Question

Numbers $1,2,3,...,2n(n\in N)$ are printed on $2n$ cards.
The probability of drawing a number $n$ is proportional to $r$. Then the probability of drawing an even number in one draw is

• A. $\frac{n+2}{n+3}$
• B. $\frac{n+1}{n+3}$
• C. $\frac{1}{2}$
• D. $\frac{n+1}{2n+1}$

Answer 1

The correct option is D $\frac{n+1}{2n+1}$

If $P\left(r\right)$ is the probability that the number $r$ is drawn in one draw, it is given that $P\left(r\right)=kr$, where $k$ is a constant.
Further $P\left(1\right)+P\left(2\right)+...+P\left(2n\right)=1$
$\Rightarrow k(1+2+3+...+2n)=1\Rightarrow k=\frac{1}{n(2n+1)}$
Hence, the required probability $=P\left(2\right)+P\left(4\right)+P\left(6\right)+...+P\left(2n\right)$
$=2k(1+2+...+n)=\frac{2}{n(2n+1)}.\frac{n(n+1)}{2}=\frac{n+1}{2n+1}$