Probability Theory

By assuming that every outcome in a sample space is equally likely, the probability of event A is then equal to the number of ways the event can occur, m, divided by the total number of outcomes that can occur, n. Symbolically, we denote the probability of event A as P(A) = m/n. An example of this is illustrated by drawing from a deck of cards. To find the probability of an event such as getting an ace when drawing a single card from a deck of cards, we must know the number of aces and the total number of cards. Each of the 4 aces represent an occupance of an event while all of the 52 cards represent the sample space. The probability of this event is then 4/52 or.08.

Using the characteristics of the sets of the sample space and an event, basic rules for probability can be created. First, since m is always equal to or less than n, the probability of any event will always be a number from 0 to 1. Second, if an event is certain to happen, its probability is 1. If it is certain not to occur, its probability is 0. Third, if two events are mutually exclusive, that is they can not occur at the same time, then the probability that either will occur is equal to the sum of their probabilities. For instance, if event A represents rolling a 6 on a die and event B represents rolling a 4, the probability that either will occur is 1/6 + 1/6 = 2/6 or 0.33. Finally, the sum of the probability that an event will occur and that it will not occur is 1.

The third rule above represents a special case of adding probabilities. In many cases, two events are not mutually exclusive. Suppose we wanted to know the probability of either picking a red card or a king. These events are not mutually exclusive because we could pick a red card that is also a king. The probability of either of these events in this case is equal to the sum of the individual probabilities minus the sum of the combined probabilities. In this example, the probability of getting a king is 4/52, the probability of getting a red card is 26/52, and the probability of getting a red king is 2/52. Therefore, the chances of drawing a red card or a king is 4/52 + 26/52 - 2/52 = 0.54.

Often the probability of one event is dependant on the occupance of another event. If we choose a person at random, the probability that they own a yacht is low. However, if we find out this person is rich, the probability would certainly be higher. Events such as these in which the probability of one event is dependant on another are known as conditional probabilities. Mathematically, if event A is dependant on another event B, then the conditional probability is denoted as P(A|B) and equal to P(A ∩ B)/P(B) when P(B) ≠ 0. Conditional probabilities are useful whenever we want to restrict our probability calculation to only those cases in which both event A and event B occur.

Events are not always dependant on each other. These independent events have the same probability regardless of whether the other event occurs. For example, probability of passing a math test is not dependent on the probability that it will rain.

Using the ideas of dependent and independent events, a rule for determining probabilities of multiple events can be developed. In general, given dependent events A and B, the probability that both events occur is P(A ∩ B) = P(B) × P(A|B). If events A and B are independent, P(A ∩ B) = P(A) × P(B). Suppose we ran an experiment in which we rolled a die and flipped a coin. These events are independent so the probability of getting a 6 and a tail would be (1/6) × 1/2 = 0.08.

The theoretical approach to determining probabilities has certain advantages; probabilities can be calculated exactly, and experiments with numerous trials are not needed. However, it depends on the classical notion that all the events in a situation are equally possible, and there are many instances in which this is not true. Predicting the weather is an example of such a situation. On any given day, it will be sunny or cloudy. By assuming every possibility is equally likely, the probability of a sunny day would then be 1/2 and clearly, this is nonsense.