**Question:**

Cards with numbers 2 to 101 are placed in a box. A card is selected at random. Find the probability that the card has

(i) an even number

(ii) a square number

**Solution:**

Total number of out comes with numbers 2 to 101, n(s) =100

(i) Let $E_{1}=$ Event of selectina a card which is an even number $=\{2,4,6, \ldots 100\}$

[in an AP, $l=a+(n-1) d$, here $l=100, a=2$ and $d=2 \Rightarrow \begin{aligned} 100=2+(n-1) 2 &\Rightarrow(n-1)=49 \Rightarrow n=50] \end{aligned}$

$\therefore$ $n\left(E_{1}\right)=50$

$\therefore$ Required probability $=\frac{n\left(E_{1}\right)}{n(S)}=\frac{50}{100}=\frac{1}{2}$

(ii) Let $E_{2}=$ Event of selecting a card which is a square number

$=\{4,9,16,25,36,49,64,81,100\}$

$=\left\{(2)^{2},(3)^{2},(4)^{2},(5)^{2},(6)^{2},(7)^{2},(8)^{2},(9)^{2},(10)^{2}\right\}$

$\therefore \quad n\left(E_{2}\right)=9$

Hence, required probability $=\frac{n\left(E_{2}\right)}{n(S)}=\frac{9}{100}$