The Variance of Random Variable X
The variance of a random variable measures how much the values of the random variable vary around the mean. It is a measure of the spread or dispersion of the random variable.

To calculate the variance, we need to know the probability distribution of the random variable. Let's assume that the random variable X has a discrete probability distribution with n possible values x1, x2, ..., xn and corresponding probabilities p1, p2, ..., pn.

Calculating the Variance
The variance of random variable X is given by the formula:

Variance(X) = E[(X - E[X])^2]

where E[X] is the expected value or mean of X.

Let's calculate the variance of random variable X step by step:

Step 1: Calculate the mean
The mean of X, denoted by E[X], is given by the formula:

E[X] = x1*p1 + x2*p2 + ... + xn*pn

Step 2: Calculate the squared difference from the mean
For each value xi, calculate the squared difference from the mean (xi - E[X])^2.

Step 3: Calculate the weighted sum
Multiply each squared difference by its corresponding probability pi, and sum them up:

Sum = (x1 - E[X])^2*p1 + (x2 - E[X])^2*p2 + ... + (xn - E[X])^2*pn

Step 4: Calculate the variance
The variance of X is obtained by dividing the sum by the total probability:

Variance(X) = Sum / 1

Applying the Formula
Let's apply the formula to the given options:
a) 1/10
b) 3/80
c) 5/16
d) 3/16

Since the question states that option 'B' is the correct answer, we need to calculate the variance using the given option.

If we consider option 'B' (3/80) as the variance, it implies that the sum of squared differences from the mean divided by 1 is equal to 3/80.

By calculating the sum of squared differences and dividing it by 1, we should obtain 3/80. If this is the case, then option 'B' is the correct answer.

However, without further information about the probability distribution of random variable X, we cannot determine the correct answer. The variance depends on the specific probabilities assigned to each value of X.