Probability theory and distributions

Probability measures how likely events are, on a scale from 0 (impossible) to 1 (certain). A random variable takes values by chance, and its distribution lists those values with their probabilities. The mean (expected value) is the long-run average, and the variance measures spread around it. Worked example: a fair die has expected value (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5, and its variance is E(X^2) - [E(X)]^2 = 91/6 - 12.25 = 15.167 - 12.25 = 2.917.