The idea of probability, chance, or randomness is quite old, whereas its rigorous axiomatization in mathematical terms occurred relatively recently. Many of the ideas of probability theory originated in the study of games of chance. In this century, the mathematical theory of probability has been applied to a wide variety of phenomena; the following are some representative examples:

• Probability theory has been used in genetics as a model for mutations and ensuing natural variability, and plays a central role in bioinformatics.

• The kinetic theory of gases has an important probabilistic component.

• In designing and analyzing computer operating systems, the lengths of various

queues in the system are modeled as random phenomena.

• There are highly developed theories that treat noise in electrical devices and com-

munication systems as random processes.

• Many models of atmospheric turbulence use concepts of probability theory.

• In operations research, the demands on inventories of goods are often modeled as

random.

• Actuarial science, which is used by insurance companies, relies heavily on the tools

of probability theory.

• Probability theory is used to study complex systems and improve their reliability,

such as in modern commercial or military aircraft.

• Probability theory is a cornerstone of the theory of finance.

A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%. A probability of 0 indicates that there is no chance that a particular event will occur, whereas a probability of 1 indicates that an event is certain to occur. A probability of 0.45 (45%) indicates that there are 45 chances out of 100 of the event occurring.

## Unconditional Probability

If we select a child at random (by simple random sampling), then each child has the same probability (equal chance) of being selected, and the probability is 1/N, where N=the population size. Thus, the probability that any child is selected is 1/5,290 = 0.0002. In most sampling situations we are generally not concerned with sampling a specific individual but instead we concern ourselves with the probability of sampling certain types of individuals. For example, what is the probability of selecting a boy or a child 7 years of age? The following formula can be used to compute probabilities of selecting individuals with specific attributes or characteristics.

**P(characteristic) = # persons with characteristic / N**

## Conditional Probability

Each of the probabilities computed in the previous section (e.g., P(boy), P(7 years of age)) is an unconditional probability, because the denominator for each is the total population size (N=5,290) reflecting the fact that everyone in the entire population is eligible to be selected. However, sometimes it is of interest to focus on a particular subset of the population (e.g., a sub-population). For example, suppose we are interested just in the girls and ask the question, what is the probability of selecting a 9 year old from the sub-population of girls? There is a total of NG=2,730 girls (here NG refers to the population of girls), and the probability of selecting a 9 year old from the sub-population of girls is written as follows:

**P(9 year old | girls) = # persons with characteristic / N**

where **| girls** indicates that we are conditioning the question to a specific subgroup, i.e., the subgroup specified to the right of the vertical line.

The conditional probability is computed using the same approach we used to compute unconditional probabilities. In this case:

**P(9 year old | girls) = 461/2,730 = 0.169.**

This also means that 16.9% of the girls are 9 years of age. Note that this is *not* the same as the probability of selecting a 9-year old girl from the overall population, which is P(girl who is 9 years of age) = 461/5,290 = 0.087.

To comprehend the theory of statistics, you must have a sound background in probability.