The variance of a random variable's probability distribution helps us to describe the distribution's:

a. shape
b. center
c. spread
d. shape, center, spread

c. spread

The correct answer is c. spread.

To understand why, let's discuss what variance represents. Variance is a measure of how spread out the values of a random variable are around the mean. It quantifies the degree of dispersion in a probability distribution.

To calculate the variance of a random variable's probability distribution, you typically need the following information:

1. The values that the random variable can take.
2. The probability of each value occurring.

The formula for variance is given by:

variance = ∑ (x - μ)^2 * P(x)

where x represents each value of the random variable, μ is the mean (or center) of the distribution, and P(x) is the probability of x occurring.

By calculating the variance, we can determine how much the values of the random variable tend to deviate from the mean. A higher variance indicates a wider spread, while a lower variance signifies a narrower spread.

Hence, the variance of a random variable's probability distribution helps describe the distribution's spread. Other characteristics such as shape and center can be described by different statistical measures, such as the skewness for shape and the mean for center.