[Math Lair] Probability

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"The laws of population growth tell us that approximately half the people who were ever born in the history of the world are now dead. There is therefore a 0.5 probability that this message is being read by a corpse."
—John W. Campbell

Probability is the study of phenomena in which chance or uncertainty plays a role. The study of probability dates to the 1650s, when the Chevalier de Méré posed two problems to his friend Blaise Pascal. The first problem was to determine the odds of throwing a 12 in 24 tosses of a pair of dice. The second problem can be summarized as follows: Two players are playing a game of chance. The first player to score three points takes the pot, and each player is equally likely to win a point. If the game is abandoned with the first player leading 2-1, how should the pot be split? (If you're interested in the answer to these questions, see the page on classical probability). Pascal was intrigued with these problems, and corresponded with Fermat about them. From the work of Pascal and Fermat, the theory of probability was born.

The probability of an event occurring can be thought of as a measurement of degree of uncertainty, or a measure of how likely an event is. Probabilities are expressed as numbers between 0 and 1, where 1 (or 100%) represents complete certainty that an event will happen, and 0 represents complete certainty that the event will not happen. Values in between 0 and 1 can be thought of as corresponding to intermediate levels of confidence that the event will occur. So, if the probability is .9 (or 90%), then the event will happen nine times out of ten; it is quite likely to happen, but not certain.

The field of probability can be seen as complementary to that of statistics. Probability theory, given information about the population, deduces probable information about a sample, while statistics, given a sample of observations, deduces information about the population as a whole.

There are three main interpretations of the concept of probability: classical probability, also known as mathematical probability or objective probability, statistical probability, also known as relative frequency probability, and subjective probability.

In classical probability, there is a sample space that consists of a number of outcomes, each of which are equally likely to occur. To determine the probability of an event occurring, simply divide the number of favourable outcomes by the total number of possible outcomes. Classical probability can be used to model events that can be repeated many times under essentially identical conditions, such as tossing coins or dice, dealing cards, and the like; in fact, probability theory was first used to describe games of chance.

Statistical probability is generally used to describe the outcome of random experiments. If you perform a certain experiment a large number of times, the probability of a certain event occurring is approximately equal to the number of successful outcomes divided by the total number of times the experiment was run.

In subjective probability, the probability of an event is estimated based on subjective factors such as personal judgement, intuition, expertise, and the like. For example, the weather forecast might predict an 80% chance of rain, based on the available information on weather patterns and what has happened on similar days.

Sources used (see bibliography page for titles corresponding to numbers): 47, 48.