Probability distributions, Expected Value and Fair Games

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

The expected value is the long-term average value of a random variable, calculated as the sum of all possible values, each multiplied by its probability.

A discrete random variable is a variable that can take on a countable number of distinct values.

A continuous random variable is a variable that can take on an infinite number of values within a given range.

Multiply each possible value by its probability and sum all the results.

P(Event) = Number of favorable outcomes / Total number of outcomes.

A game is considered fair if all players have an equal chance of winning, meaning the expected value of the game is zero.

The complement of an event is the set of all outcomes in which the event does not occur.

P(at least one) = 1 - P(none).

The sum of all probabilities in a probability distribution must equal 1.