# Properties of probability and different types of probabilities

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Properties of probability and different types of probabilities

The probability of an event is defined as the chances that the stated event will occur. It is a number between 0 and 1. If the probability of an event is 0, then the event has no chance of occurrence. If the probability of an event is one, then it is certain that the event will occur.

There are two defining properties of an event:

(1) The probability of any event is a number between 0 and 1.
(2) The sum of probabilities of any set of mutually exclusive and exhaustive events equals 1.

The sum of probabilities of all exhaustive events is greater than or equal to one. The sum of probabilities of all mutually exclusive events is less than or equal to one.

The estimation of the probability of an event using experiments or based on the frequency of occurrence of the event historically is known as empirical probability. For example, if the annual returns of an index have been positive 15 times out of 20 times, then the empirical probability of an event that the index earns a positive annual return is 75 percent. (=15/20 = 0.75)

A subjective probability is estimated using subjective judgment. It is of great importance in investments as all investors make buying and selling decisions based on their subjective probabilities of the movement in the stock price. For example, if an investor thinks that the probability of the index fund earning a positive return is 80 percent then that is a subjective probability. The market price of the index is determined by the interaction of such subjective probabilities of all the investors in the market.

A priori probability is an estimation of the probability of an event using logical analysis or some mathematical rules. For example, when you throw a fair dice, then the probability of getting any number is equal to 1/6. It is logical because all six outcomes are equally likely. The empirical probability and a priori probability do not vary from person to person. Hence, they are also called as objective probabilities.

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