Introduction to Probability: Definition of Terms

Probability is a measure of the likelihood that something will happen. We usually express probabilities as percentages, from 0 (impossible to happen) to 100% (guaranteed). In fact, we can express almost any event as a probability, e.g., the likelihood that it will rain on a given day or the likelihood of one passing an examination.

Estimation of probabilities is imperative in financial management. More precisely, concepts in probability help analysts quantify risks. The following are the major probability concepts you should know.

Random Variable

A random variable is any quantity whose expected future value is not known in advance. Examples include the expected future value of a mutual fund, expected dividend payment on a stock, or the expected standard deviation of investment returns. We cannot know the exact future value of an investment since it is usually a function of multiple factors, some of which may be beyond the control of the financial manager.

Note that returns that are fixed cannot be described as random variables. For instance, if a government bond is quoted at a fixed discount of, say, 6%, the bond’s future value can be calculated in advance. Therefore, it is not a random variable.

Outcome

An outcome is any possible value that a random variable can take. For example, if you roll a six-sided dice, there are six possible outcomes since any number, from 1 to 6, can come up.

If, for example, 3 comes up, then that is an outcome. In case 5 comes up, it’s another outcome. So, for example, if a stock offers shareholders a $2 dividend per share at the end of a year, then the $2 dividend is an outcome.

Event

A single outcome or a set of outcomes is known as an event. If we take the dice example above, rolling a 4 is an event, rolling an odd number.

Mutually Exclusive Events

Mutually exclusive events are such that one event precludes the occurrence of all the other events. Thus, if you roll a dice and a 4 comes up, that particular event precludes all the other events, i.e., 1, 2, 3, 5, and 6. In other words, rolling a 1 and a 5 are mutually exclusive events: they cannot occur simultaneously.

Furthermore, there is no way a single investment can have more than one arithmetic mean return. Therefore, arithmetic returns of, say, 20% and 17% constitute mutually exclusive events.

Exhaustive Events

Events are said to be exhaustive if they include all possible outcomes. Suppose we roll a dice once and categorize the outcomes into two events as follows:

{1, 3, 5}   {2, 4, 6}

Each of the above set of outcomes is an event. The two events are exhaustive because they include all the possible outcomes.