Independence and Expectation

Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a fair coin twice: the result of the first flip doesn't change the odds for the second. In contrast, dependent events influence each other, such as drawing cards without replacement.

Expected value (or mean) is a central concept in probability, representing the long-run average outcome of a random process.

It is calculated as \( E(X) = \sum x \times P(X = x) \) for discrete variables.

It has useful properties such as linearity: \( E(aX + b) = aE(X) + b \), and additivity: \( E(X + Y) = E(X) + E(Y) \).