Question

In a lottery of $50$ tickets numbered $1$ to $50$, two tickets are drawn simultaneously. The probability that both the tickets are drawn have prime numbers is ?

Answer 1

Step 1: Find the probability of getting a prime number in the first draw

Given: In a lottery of $50$ tickets numbered $1$ to $50$, two tickets are drawn simultaneously.

We know that the number of prime numbers between $1$ to $50$ is $15\left\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\right\}$.

Probability is given as

$\mathbf{Probability}\mathbf{=}\frac{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{favourable}\mathbf{}\mathbf{outcomes}}{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{total}\mathbf{}\mathbf{outcomes}}$

So, the probability of getting a prime number in the first draw will be$=\frac{\mathrm{Total}\mathrm{prime}\mathrm{number}\mathrm{between}1\mathrm{to}50}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tickets}}$

$=\frac{15}{50}$

Step 2: Find the probability of getting a prime number in the second draw

If $1$ card is taken from the pack of lottery tickets, then the number of remaining prime numbers are $14$.

The number of remaining lottery tickets $=49$

So, the probability of getting a prime number in the second draw will be$=\frac{\mathrm{Remaining}\mathrm{prime}\mathrm{numbers}}{\mathrm{Remaning}\mathrm{number}\mathrm{of}\mathrm{tickets}}$

$=\frac{14}{49}$

Therefore,

Step 3: Find the combined probability of getting $2$ prime numbers in $2$ draws

the probability of getting prime numbers in both draws$=$ (The probability of getting a prime number in the first draw) $\times$(The probability of getting a prime number in the second draw )

$$\begin{gathered} =\left(\frac{15}{50}\right)\times \left(\frac{14}{49}\right) \\ =\frac{210}{2450} \\ =\frac{3}{35} \end{gathered}$$

Hence, the probability that both the tickets are drawn have prime numbers is $\frac{3}{35}$.