LOS A, B, and C require us to:
a. define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events,
b. state the two defining properties of probability and distinguish among empirical, subjective, and a priori probabilities; and
c. state the probability of an event in terms of odds for and against the event
a. Random Variables: Random variables are a quantity whose outcomes or possible values are unknown.
For example, the expected return of 10% from an investment is only an expectation of the manager, it is not certain.
b. Event: Event is a specified set of outcomes. It could be either a point or a range of different points. It is usually denoted by the letters A, B, C, D, and so on.
For example, a return earned of 10% is a point event and all the returns above 8% are the range of events.
c. Probability of an Event: The probability of an event is the possibility of its occurrence. It is denoted by P(A).
The probability of any event always lies between 0 and 1. This is mainly because we cannot have either a negative probability or the probability greater than 100%. Thus:
Also, the sum of possibilities of all the events is always equal to 1 or 100%. That is:
These two conditions are true for any set of
i. mutually exclusive events (i.e. those events that cannot take place simultaneously) or
ii. exhaustive events (that covers all the possible events).
Therefore, in order to calculate the probability of an event we need two things:
iii. set of all distinct possible outcomes, and
iv. the probability distribution.
d. Empirical Probability: It is the probability of an event as a relative frequency of occurrence based on historical data. Thus, for the empirical probability:
i. The past is assumed to be representative of the future, and
ii. The historical period must include the occurrences of the event.
e. Subjective Probability: Subjective probability is the one that is subject to the personal judgment of the analyst. The subjective probabilities can be assigned to an event:
i. by adjusting an empirical probability based on intuition or experience,
ii. when there is a complete lack of empirical observations, and
iii. to make a personal assessment.
f. A Priori Probability: An a priori probability is used when there is neither historical evidence nor knowledge or experience to derive the subjective probability. An a priori probability is based on deductive reasoning.
f. Odds: Odds of an event refers to the chances of their occurrence or nonoccurrence.
The odds for an event can be calculated through this equation:
So, for example, if the probability of an event is 0.10, the odds for the event would be:
or 1 to 9, i.e. for each occurrence of event E, we can expect 9 events of its non-occurrence. Thus, the odds for an event are expressed as ‘a to b’, meaning that for each ‘a’ chance of occurrence of an event, we can expect ‘b’ chances of its nonoccurrence.
The odds against an event can be calculated through this equation:
So, for example, if the probability of an event is 0.10, the odds for the event would be:
or 9 to 1, i.e. for every 9 occurrences of event E, we can expect 1 event of its non-occurrence. Thus, the odds for an event are expressed as ‘b to a’, meaning that for each ‘a’ chance of non-occurrence of an event, we can expect ‘b’ chances of its occurrence.