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(C) Let $S = \{1, 2, \dots, 11\}$. The set contains $5$ even numbers $\{2, 4, 6, 8, 10\}$ and $6$ odd numbers $\{1, 3, 5, 7, 9, 11\}$.The sum of two n

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$\frac{2}{5}$

Vedclass · Answer

(C) Let $S = \{1, 2, \dots, 11\}$. The set contains $5$ even numbers $\{2, 4, 6, 8, 10\}$ and $6$ odd numbers $\{1, 3, 5, 7, 9, 11\}$.The sum of two numbers is even if both are even or both are odd.Number of ways to select two even numbers: $^5C_2 = \frac{5 	imes 4}{2} = 10$.Number of ways to select two odd numbers: $^6C_2 = \frac{6 	imes 5}{2} = 15$.Total ways to get an even sum: $10 + 15 = 25$.The conditional probability that both are even given the sum is even is:$P = \frac{	ext{Number of ways both are even}}{	ext{Total ways sum is even}} = \frac{10}{25} = \frac{2}{5}$.

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

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