# Probability Topics

Probability is the measure of the likelihood that an event will occur. It is quantified as the ratio of the number of favorable outcomes (outcomes that are true according to the laws of the probability theory) to the total number of possible outcomes. The likelihood of an event is determined by the number of favorable outcomes multiplied by the probability of each outcome. An event is said to be improbable if there is a small number of favorable outcomes. The mathematical concept of probability can be defined in a variety of ways. Most simply, it is the degree of belief that some outcome will occur. This is the definition that is used most frequently. Probability is a central concept in any discussion of random events. Probability theory is the mathematical branch of applied mathematics devoted to the analysis of random phenomena. Probability theory provides the mathematical underpinnings for the interpretation of experimental results in science, and has provided insights into the workings of nature. Its earliest known uses are in games of chance, but it is now applied in a wide variety of areas of science and in everyday life. For instance, the probability that a person at a casino will win a jackpot increases as the size of the jackpot increases, unless they hit a series of jackpots (see martingale strategy), but the probability that a person will win in a game of chance is not enough to make it an even bet. Probability theory is used in the sciences to describe the behavior of physical systems, such as the motion of objects, the results of statistical experiments, and the results of stochastic processes, such as the outcomes of rolls of a die. Probability theory can also be used to develop new mathematical theories in other fields, such as economics and linguistics. In areas of human activity where correct decisions have large consequences, such as in determining the success of a venture or military strength, probability theory is important. Probability theory is used to understand the world in a wide variety of ways. Probability theory is used in the social sciences to describe the behavior of individuals in situations involving risk, and is also used to understand the behavior of non-human systems, such as the weather. Probability theory is also used in science to interpret the results of scientific experiments, in engineering applications, in game theory, and in many other areas. The modern theory of probability originated with the work of Pierre de Fermat. Fermat developed a study of probability by asking the question: "Given that an event has occurred, what is the probability that another event also has occurred?" For example, if a fair die is rolled, the probability of each of the six possible outcomes, such as "1", "2", "3", "4", "5", or "6", is one-sixth. This leads to the following definition of probability: P(A) = the probability of A This definition can be read as: the probability of A is the number of favorable outcomes of A divided by the total number of possible outcomes. The total number of possible outcomes is the number of ways in which the outcome of the experiment can be, or can be of the form (i.e. a probability): P(A) = the number of favorable outcomes of A The number of favorable outcomes is the number of outcomes which are true, or favorable to the occurrence of A, given that the experiment has been performed: P(A) = the number of favorable outcomes This definition can also be read as: the probability of A is the number of favorable outcomes of A divided by the total number of possible outcomes. The problem of probability is to assign numbers to the possible events, and then to find the probability of each event. To do this, we must know the possible events, their probability, and the number of outcomes. We can then use Bayes' theorem to find the probability of every outcome. Given a single possible event, such as a coin toss, there are only two possible outcomes, head or tail. The probability of head is expressed as , and the probability of tail as . For two events, such as the toss of two coins, there are four possible outcomes. The probability of the first event, such as the first coin landing heads, is expressed as , and the probability of the second event, such as the second coin landing heads, is expressed as . When we toss a coin twice, there are four possible outcomes, heads or tails on both coins, and the probability of both coins landing heads is expressed as . Given that a fair coin has a probability of one-half of landing heads and one-half landing tails, the probability of a coin landing head is expressed as . The probability of a coin landing tail is expressed as . Given that a die has six sides, there are six possible outcomes, which can