# A box contains cards numbered 3, 5, 7, 9, ..., 35, 37. A card is drawn at random form the box.

Question:

A box contains cards numbered 3, 5, 7, 9, ..., 35, 37. A card is drawn at random form the box. Find the probability that the number on the drawn card is a prime number.

Solution:

The numbers 3, 5, 7, 9, ..., 35, 37 are in AP.

Here, a = 3 and d = 5 − 3 = 2

Suppose there are n terms in the AP.

$\therefore a_{n}=37$

$\Rightarrow 3+(n-1) \times 2=37 \quad\left[a_{n}=a+(n-1) d\right]$

$\Rightarrow 2 n+1=37$

$\Rightarrow 2 n=37-1=36$

$\Rightarrow n=18$

∴ Total number of outcomes = 18

Let E be the event of drawing a card with prime number on it.

Out of the given numbers, the prime numbers are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and 37.

So, the favourable number of outcomes are 11.

$\therefore$ Required probability $=\mathrm{P}(\mathrm{E})=\frac{\text { Favourable number of outcomes }}{\text { Total number of outcomes }}=\frac{11}{18}$