Probability refers to the likelihood or chance that some uncertain outcome will occur. In probability theory, each occurrence whose outcome we are uncertain about is called an event, while the set of all possible outcomes of the event is called the sample space. Each possible outcome in the sample space is assigned a number between 0 and 1 called the probability of that outcome, which describes how likely that outcome is to occur. A value of 0 indicates that the outcome is impossible, while a value of 1 indicates that the outcome is certain. The total probability of all possible outcomes must always add up to 1, since some outcome is certain to occur.

Simple examples of coin tossing and dice throwing are often used to illustrate the key concepts of probability. In the case of tossing a fair coin on a flat surface, the uncertain event is what side the coin will land on, since we don’t know in advance what this will be. The sample space, the set of all possible outcomes, consists of two elements: heads and tails. Assuming the coin is fair, each has an equal chance of occurring, and thus has a probability of 0.5. In the case of rolling a six-sided die, the set of possible outcomes is {1,2,3,4,5,6}, each with an equal 1/6 chance of occurring.

Many phenomena in science, politics, society, and even philosophy, are uncertain, and thus can be usefully described in probabilistic terms. One can consider the probability of a certain political candidate winning an election, of a given country winning a particular war, of a certain DNA match occurring by chance, of an atomic nucleus decaying in a certain period of time, of the stock market going up or down by a certain number of points, and so on. The ubiquitous presence of probability in so many facets modern life means that having some basic familiarity with the basic concepts of probability is vital to being an informed citizen and a capable critical thinker.

Further Reading

LessWrong probability resources: a set of links and resources on probability and statistics

Notes on probability: set of notes on basic concepts in probability (pdf)

Basic concepts in probability: a short introduction