Question

Two integers are selected at random from the set $\{1,2,\dots ,11\}.$ Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

• A. $\frac{2}{5}$
• B. $\frac{1}{2}$
• C. $\frac{3}{5}$
• D. $\frac{7}{10}$

Answer 1

The correct option is A $\frac{2}{5}$

Since, it is given that the sum of the selected numbers is even therefore, we can select either both odd numbers or both even numbers.
For sum of the selected numbers to be even, number of ways are ${}^5{C}_2+{}^6{C}_2$
$\therefore$ Required conditional probability
$=\frac{{}^5{C}_2}{{}^5{C}_2+{}^6{C}_2}$
$=\frac{\frac{5\times 4}{2}}{\frac{5\times 4}{2}+\frac{6\times 5}{2}}=\frac{2}{5}$