Question

Out of $30$ consecutive numbers, $2$ are chosen at random. The probability that their sum is odd is

• A. $\frac{14}{29}$
• B. $\frac{16}{29}$
• C. $\frac{15}{29}$
• D. $\frac{10}{29}$

Answer 1

The correct option is C

$\frac{15}{29}$

Explanation for correct option

Calculating the probability of selecting two random numbers whose sum is odd:

Given, there are $30$ consecutive numbers.

Total cases of selecting any two random numbers $={}^{30}C_2=435$

The sum is odd, if one of the integers is odd and the other integer is even.

If there are $30$ consecutive numbers then there will be $15$ odd numbers and $15$ even numbers.

So, the favourable cases $={}^{15}C_1\times {}^{15}C_1=225$

Therefore, the required probability of selecting two random numbers such that their sum is odd is:

$$\begin{gathered} =\frac{\mathrm{Favourable}\mathrm{ways}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{ways}} \\ =\frac{15\times 15}{435} \\ =\frac{225}{435} \\ =\frac{15}{29} \end{gathered}$$

Hence, the correct answer is Option (C).