# Discovery Science: Probability – Probability Theory

## Probability – Probability Theory

Probability theory, the mathematical study of the estimated consequences of future events, is useful for predicting costs and planning complex processes when not all variables are available.

If the consequences of a future event have to be estimated, probability theory is used to help make an informed decision. For example, a company insuring for accidents must cover the risk to the insurer, without high costs. If the risk of an accident occurring over one year is estimated at five percent, the yearly premium will therefore be at least five percent of the average cost of an accident.

Probability theory was developed from questions about chance in gambling. The probability of an event is calculated by dividing the number of specific outcomes in that event by the number of all possible outcomes. For example, the probability of drawing a black card from a deck of cards is 0.50, or 50 percent (26 black cards out of 52 cards total). If the outcomes are part of subevents, a decision must also be made as to whether they are dependent on each other.

For instance, if you begin drawing cards from a single deck, the chance that you will draw a queen of hearts increases as the size of the deck decreases (1/52, 1/51, 1/50, and so on). To find the probability of several independent events occurring, you must multiply the individual probabilities.

For example, the chance that you would roll two dice and both would come up “six” is P(X=6) =1/36 or 2.8 percent. Probability theory is particularly useful for making responsible decisions in economic or social situations.

Limits of probability theory

There are some probabilities that cannot be calculated accurately. For example, when a person is faced with a choice be- tween plain chocolate and chocolate with peanuts, the probability that the plain chocolate is chosen could be calculated at 0.5. However, other factors, such as an allergy to peanuts, may influence the decision.

### STOCHASTIC THEORY

Stochastic theory includes the fields of probability and statistics. It concerns itself with situations which are not fully predictable such as economic developments, quantum mechanics, changes in animal populations, the effects of medications, and failure rates for mechanical equipment, in addition to natural weather predictions.

All of these models must include parameters that can be adjusted. Stochastic theory is used to help researchers choose the most appropriate parameters and test their models

### THE GALTON BOX

A device developed in the 19th century by Sir Francis Galton (1822— 1911) provides a model for experiments involving two options in a series of repetitions. It consists of a vertical board with a grid of pins. Balls dropped into the grid at its center strike the pins and have an equal probability of falling to the left or right.

With eight pins, there are 28 = 256 possible paths, most of which lead to the central tracks. The binomial distribution can be used to calculate the individual probabilities,