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# Two integers are selected at random from the set

Question:

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

1. $\frac{2}{5}$

2. $\frac{1}{2}$

3. $\frac{3}{5}$

4. $\frac{7}{10}$

Correct Option: 1

Solution:

Since sum of two numbers is even so either both are odd or both are even. Hence number of elements in reduced samples space

$={ }^{5} C_{2}+{ }^{6} C_{2}$

so required probability $=\frac{{ }^{5} \mathrm{C}_{2}}{{ }^{5} \mathrm{C}_{2}+{ }^{6} \mathrm{C}_{2}}$