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A probability distribution is a list or a table that shows all the possible values of a random variable and the likelihood, or probability, that each value will occur. In our example, the random variable \( x \) can take on the values 0, 1, 2, and 3. Each of these values has an associated probability \( P(x) \) which tells us how likely that specific outcome is. For example, the chance that \( x \) is 2 is known to be 0.4, or 40%.
Understanding probability distributions is crucial because they allow us to predict the likelihood of different outcomes in random events. By organizing these probabilities in a table, we can perform further statistical calculations, like finding the expected value or variance. Remember, all probabilities in a distribution must add up to 1, because one of the possible outcomes must occur. In this case, \( 0.2 + 0.2 + 0.4 + 0.2 = 1.0 \).
When reading a probability distribution table, note how probabilities reflect the chance of observing each outcome. This is instrumental in making predictions and informed decisions in real-life situations or statistical analyses.

A random variable is a numerical outcome of a random process or experiment. It represents possible values from a random event along with accompanying probabilities from a probability distribution. In our exercise, \( x \) is the random variable representing different outcomes based on the context of the problem or experiment that is not specified. For instance, \( x = 0 \) could mean 0 successes in a series of trials, whereas \( x = 3 \) might mean 3 successes.
Random variables can be either discrete or continuous. Discrete random variables, like our \( x \), have distinct, separate values, such as 0, 1, 2, or 3. Meanwhile, continuous random variables can take any value within a specified range. It's important to realize that random variables themselves are not random; their outcomes or values are determined by underlying randomness.
This concept is key in probability and statistics as it helps us model and analyze real-world situations quantitatively. Knowing how to handle random variables enables interpretations of complex systems in fields like finance, engineering, and social sciences.

Probability calculation is the process we perform to determine the probability of a particular outcome of a random variable. In this exercise, we used a specific type of probability calculation known as the expected value. It involves multiplying each outcome by its probability and summing these products.
To complete our table, we calculated \( x^* P(x) \) for each value of \( x \):

• For \( x = 0 \), \( x^* P(x) = 0 \times 0.2 = 0 \)
• For \( x = 1 \), \( x^* P(x) = 1 \times 0.2 = 0.2 \)
• For \( x = 2 \), \( x^* P(x) = 2 \times 0.4 = 0.8 \)
• For \( x = 3 \), \( x^* P(x) = 3 \times 0.2 = 0.6 \)

These calculations help us understand the average or expected value of \( x \) over many trials. The expected value gives a sense of the 'center' of a probability distribution and is a single summary measure that can describe a random variable's likely behavior over time. Thus, probability calculations are foundational in predicting and interpreting random phenomena.