SATHEE: Probability 3 Question 4

Probability 3 Question 4

4. 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

(2019 Main, 11 Jan I)

(a) $\frac{2}{5}$

(b) $\frac{1}{2}$

(d) $\frac{3}{5}$

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Solution:

In $1, 2, 3, \dots ., 11$ there are 5 even numbers and 6 odd numbers. The sum even is possible only when both are odd or both are even.

Let $A$ be the event that denotes both numbers are even and $B$ be the event that denotes sum of numbers is even. Then, $n (A) =^{5} C_{2}$ and $n (B) =^{5} C_{2} +^{6} C_{2}$

Required probability

$\begin{aligned} P (A / B) = \frac{P (A \cap B)}{P (B)} & = \frac{^{5} C_{2} /^{11} C_{2}}{\frac{(^{6} C_{2} +^{5} C_{2})}{^{11} C_{2}}} \\ = \frac{^{5} C_{2}}{^{6} C_{2} +^{5} C_{2}} = \frac{10}{15 + 10} = \frac{2}{5} \end{aligned}$

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Probability 3 Question 4

4. Two integers are selected at random from the set 1,2$,$…...,11. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is

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4. 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