# Defining Properties of Probability

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

### The Probability of an Event is Always a Number Between 0 and 1, Inclusively

That is,

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

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

### The Sum of All Probabilities of All Events = 1, Provided the Events are Mutually Exclusive and Exhaustive

That is:

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

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

#### Example: Exhaustive Events

Suppose we toss a fair coin. The only possible outcome is either a head or a tail. Either outcome has a probability of 0.5. Therefore,

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

Obtaining a head 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.

#### Example: Mutually Exclusive Events

Which of the following events are mutually exclusive?

1. Drawing a ”ten” card and drawing a King from a well-shuffled standard deck of playing cards.
2. Drawing a black card and drawing a club from a well-shuffled standard deck of playing cards.

Solution

1. Since A = {ten of diamonds, ten of hearts, ten of clubs, ten of spades} and B = {king of diamonds, king of hearts, king of clubs, king of spades}, then A and B are mutually exclusive since there are no cards common to both events.
2. Drawing a black card and drawing a club are not mutually exclusive events because all the clubs are black.

## Empirical Probability

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

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

## Subjective Probabilities

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

This approach is subject to personal flaws and talents. For this reason, 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 either from subjective judgment or an empirical approach. Let us 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.

## Question

An analyst is analyzing the preference of retired individual investors with respect to investing their pension savings. He surveyed 80 investors and observed that 70 out of 80 people chose to invest their pension savings in money market funds instead of equity funds. What is the empirical probability of any retired individual investing in an equity fund?

1. 13%.
2. 75%.
3. 87%.

Solution:

$$\text{Empirical Probability} = \frac{10}{80} = 13\%$$.

The empirical probability of any retired individual selecting equity fund for retirement saving investment is 13%.

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