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

Answer: (D)

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