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Math probabilities

Notes from school 

Encarta notes 

, also theory of probability, branch of mathematics that deals with measuring or determining quantitatively the likelihood that an event or experiment will have a particular outcome. Probability is based on the study of permutations and combinations and is the necessary foundation for statistics.

The foundation of probability is usually ascribed to the 17th-century French mathematicians Blaise Pascal and Pierre de Fermat, but mathematicians as early as Gerolamo Cardano had made important contributions to its development. Mathematical probability began in an attempt to answer certain questions arising in games of chance, such as how many times a pair of dice must be thrown before the chance that a six will appear is 50-50. Or, in another example, if two players of equal ability, in a match to be won by the first to win ten games, are obliged to suspend play when one player has won five games, and the other seven, how should the stakes be divided?

The probability of an outcome is represented by a number between 0 and 1, inclusive, with "probability 0" indicating certainty that an event will not occur and "probability 1" indicating certainty that it will occur. The simplest problems are concerned with the probability of a specified "favorable" result of an event that has a finite number of equally likely outcomes. If an event has n equally likely outcomes and f of them are termed favorable, the probability, p, of a favorable outcome is f/n. For example, a fair die can be cast in six equally likely ways; therefore, the probability of throwing a 5 or a 6 is 2/6. More involved problems are concerned with events in which the various possible outcomes are not equally likely. For example, in finding the probability of throwing a 5 or 6 with a pair of dice, the various outcomes (2, 3, ... 12) are not all equally likely. Some events may have infinitely many outcomes, such as the probability that a chord drawn at random in a circle will be longer than the radius.

Problems involving repeated trials form one of the connections between probability and statistics. To illustrate, what is the probability that exactly five 3s and at least four 6s will occur in 50 tosses of a fair die? Or, a person, tossing a fair coin twice, takes a step to the north, east, south, or west, according to whether the coin falls head, head; head, tail; tail, head; or tail, tail. What is the probability that at the end of 50 steps the person will be within 10 steps of the starting point?

In probability problems, two outcomes of an event are mutually exclusive if the probability of their joint occurrence is zero; two outcomes are independent if the probability of their joint occurrence is given as the product of the probability of their separate occurrences. Two outcomes are mutually exclusive if the occurrence of one precludes the occurrence of the other; two outcomes are independent if the occurrence or nonoccurrence of one does not alter the probability that the other will or will not occur. Compound probability is the probability of all outcomes of a certain set occurring jointly; total probability is the probability that at least one of a certain set of outcomes will occur. Conditional probability is the probability of an outcome when it is known that some other outcome has occurred or will occur.

If the probability that an outcome will occur is p, the probability that it will not occur is q = 1 - p. The odds in favor of the occurrence are given by the ratio p:q, and the odds against the occurrence are given by the ratio q:p. If the probabilities of two mutually exclusive outcomes X and Y are p and P, respectively, the odds in favor of X and against Y are p to P. If an event must result in one of the mutually exclusive outcomes O₁,O₂,..., O_n, with probabilities p₁,p₂,..., p_n, respectively, and if v₁,v₂,...v_n are numerical values attached to the respective outcomes, the expectation of the event is E = p₁v₁ + p₂v₂ + ...p_nv_n. For example, a person throws a die and wins 40 cents if it falls 1, 2, or 3; 30 cents for 4 or 5; but loses $1.20 if it falls 6. The expectation on a single throw is 3/6 × .40 + 2/6 × .30 - 1/6 × 1.20 = .10.

The most common interpretation of probability is used in statistical analysis. For example, the probability of throwing a 7 in one throw of two dice is 1/6, and this answer is interpreted to mean that if two fair dice are randomly thrown a very large number of times, about one-sixth of the throws will be 7s. This concept is frequently used to statistically determine the probability of an outcome that cannot readily be tested or is impossible to obtain. Thus, if long-range statistics show that out of every 100 people between 20 and 30 years of age, 42 will be alive at age 70, the assumption is that a person between those ages has a 42 percent probability of surviving to the age of 70.

Mathematical probability is widely used in the physical, biological, and social sciences and in industry and commerce. It is applied in such diverse areas as genetics, quantum mechanics, and insurance. It also involves deep and important theoretical problems in pure mathematics and has strong connections with the theory, known as mathematical analysis, that developed out of calculus.

Possible Outcomes

Outcome: the nature of possibilities that may occur.

e.g. tossing a coin has 2 possible outcomes, which are the appearance of heads or tails

Event: the occurrence of any one of the possible outcomes

e.g. If a coin is tossed 10 times 10 is the total number of times the event takes place.

Frequency: the number of times that an outcome occurs

e.g. number of times coin lands with heads, and number of times coin lands with tails.

Three diagrams: lists the possible outcome of an experiment.

e.g. coin toss: probability of coin landing head or tail.

Equally: likely outcomes that have the same chance of occurring.

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Last modified: May 21, 2001