Algebra

Probability

Easy

core

60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

$\frac{3}{5}$

$\frac{6}{5}$

$\frac{6}{25}$

$\frac{3}{10}$

✅ Correct Option: 3

Given that 60% of committee members favor the proposal and 40% oppose it. A member is selected with random variable $X = 1$ if favoring and $X = 0$ if opposing.

The probabilities are:

$P(X = 1) = 0.6 = \frac{3}{5}$

$P(X = 0) = 0.4 = \frac{2}{5}$

The expected value is:

$E(X) = 0 \times P(X=0) + 1 \times P(X=1)$

$E(X) = 0 \times \frac{2}{5} + 1 \times \frac{3}{5}$

$E(X) = \frac{3}{5}$

The expected value of $X^2$ is:

$E(X^2) = 0^2 \times P(X=0) + 1^2 \times P(X=1)$

$E(X^2) = 0 \times \frac{2}{5} + 1 \times \frac{3}{5}$

$E(X^2) = \frac{3}{5}$

The variance is:

$\text{Var}(X) = E(X^2) - [E(X)]^2$

$\text{Var}(X) = \frac{3}{5} - \left(\frac{3}{5}\right)^2$

$\text{Var}(X) = \frac{3}{5} - \frac{9}{25}$

$\text{Var}(X) = \frac{15}{25} - \frac{9}{25}$

$\text{Var}(X) = \frac{6}{25}$

Therefore, the variance of the random variable $X$ is $\frac{6}{25}$ .

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60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable X = 0 if he opposes and X = 1 if he is in favour. Then the variance of the random variable X is

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