Defining Properties of Probability

Defining properties of a probability refer to the rules that constitute any given probability. These are:

(I) The probability of an event is always a number between 0 and 1, inclusively.

$$ 0 \le P(E_i) \le 1 \quad \quad  \text{for i} = 1,2…n $$

A ‘P’ followed by E_i in parentheses is interpreted as the probability of an event E_i. We cannot have a negative probability or probability greater than 1 (100%). There is nothing like more certain than certain!

(II) The sum of all probabilities of all events = 1, provided the events are mutually exclusive and exhaustive.

$$ \sum { P\left( { E }_{ i } \right) =1 } \quad \quad \text{for i} = 1,2…n $$

Candidates should note that if the events are not mutually exclusive, the total probability would be greater than 1. Similarly, if the events are not exhaustive, the total probability would be less than 1 (some events would be left out).

Example

Suppose we toss a fair coin. The only possible outcomes are either ahead or a tail. Each outcome has a probability of 0.5. Therefore,

$$ P(H) + P(T) = 0.5 + 0.5 = 1 $$

Obtaining ahead precludes obtaining a tail. Thus the two events are mutually exclusive. Similarly, there is no other possible outcome apart from a head or a tail. Therefore, the two events are exhaustive.

Empirical Probability

An empirical probability is a probability that results from analyzing actual past data. For example, if we assembled the returns earned by a stock for the last 25 years and used them to make future forecasts, then we have employed an empirical approach.

One drawback about empirical probabilities is that they rely on past performance, which is not always indicative of future performance. Certain events could occur in the future, leading to drastic changes in returns.

Subjective Probabilities

Subjective probabilities usually reflect personal belief or judgment. Thus, analysts may rely on their personal experience and judgment when estimating future performance.

This approach is subject to personal flaws and talents. Therefore, the probabilities churned out may not be very accurate and are likely to differ, even among fund managers working for the same company.

A Priori Probabilities

A priori probabilities are subjective, deductive, and based on reasoning. For example, suppose we establish that a fund manager has an 80% chance of securing a new job in a certain company. The 80% probability could have resulted from either subjective judgment or an empirical approach. Let’s assume that the fund manager has only one competitor. If we apply deductive reasoning in this scenario, then we would conclude that the competitor has a 20% chance of securing the job.

Reading 8 LOS 3b: identify the two defining properties of probability, including mutually exclusive and exhaustive events, and compare and contrast empirical, subjective, and a priori probabilities;

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