Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then the variance of $x$ is

Step 1: Identify the total number of oranges and defective oranges

First, we identify the total number of oranges available, which is the sum of defective and good oranges. We also note the number of defective oranges and good oranges separately.

Step 2: Determine possible values of x and their probabilities

Let $x$ be the number of defective oranges drawn. Since two oranges are drawn, $x$ can take values 0, 1, or 2. We calculate the probability for each value using combinations. The total number of ways to draw 2 oranges from 10 is $\binom{10}{2} = 45$.

Step 3: Calculate the expected value of x, E(x)

The expected value of $x$, denoted as $E(x)$, is calculated by summing the product of each possible value of $x$ and its corresponding probability.

Step 4: Calculate the expected value of x squared, E($x^2$)

Next, we calculate the expected value of $x^2$, denoted as $E(x^2)$, by summing the product of the square of each possible value of $x$ and its corresponding probability.

Step 5: Calculate the variance of x

Finally, we calculate the variance of $x$ using the formula $\text{Var}(x) = E(x^2) - (E(x))^2$. We substitute the values we calculated for $E(x^2)$ and $E(x)$ and simplify the expression.

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