To help make logical and consistent investment decisions, and help manage expectations in an environment of risk, an analyst uses the concepts and tools found in probability theory. A probability refers to the percentage chance that something will happen, from 0 (it is impossible) to 1 (it is certain to occur), and the scale going from less likely to more likely. Probability concepts help define risk by quantifying the prospects for unintended and negative outcomes; thus probability concepts are a major focus of the CFA curriculum.

I. Basics

Random Variable
A random variable refers to any quantity with uncertain expected future values. For example, time is not a random variable since we know that tomorrow will have 24 hours, the month of January will have 31 days and so on. However, the expected rate of return on a mutual fund and the expected standard deviation of those returns are random variables. We attempt to forecast these random variables based on past history and on our forecast for the economy and interest rates, but we cannot say for certain what the variables will be in the future - all we have are forecasts or expectations.

Outcome
Outcome refers to any possible value that a random variable can take. For expected rate of return, the range of outcomes naturally depends on the particular investment or proposition. Lottery players have a near-certain probability of losing all of their investment (-100% return), with a very small chance of becoming a multimillionaire (+1,000,000% return - or higher!). Thus for a lottery ticket, there are usually just two extreme outcomes. Mutual funds that invest primarily in blue chip stocks will involve a much narrower series of outcomes and a distribution of possibilities around a specific mean expectation. When a particular outcome or a series of outcomes are defined, it is referred to as an event. If our goal for the blue chip mutual fund is to produce a minimum 8% return every year on average, and we want to assess the chances that our goal will not be met, our event is defined as average annual returns below 8%. We use probability concepts to ask what the chances are that our event will take place.

Event
If a list of events ismutually exclusive, it means that only one of them can possibly take place. Exhaustive events refer to the need to incorporate all potential outcomes in the defined events. For return expectations, if we define our two events as annual returns equal to or greater than 8% and annual returns equal to or less than 8%, these two events would not meet the definition of mutually exclusive since a return of exactly 8% falls into both categories. If our defined two events were annual returns less than 8% and annual returns greater than 8%, we've covered all outcomes except for the possibility of an 8% return; thus our events are not exhaustive.

The Defining Properties of Probability
Probability has two defining properties:
1. The probability of any event is a number between 0 and 1, or 0 < P(E) < 1. A P followed by parentheses is the probability of (event E) occurring. Probabilities fall on a scale between 0, or 0%, (impossible) and 1, or 100%, (certain). There is no such thing as a negative probability (less than impossible?) or a probability greater than 1 (more certain than certain?).

2. The sum of all probabilities of all events equals 1, provided the events are both mutually exclusive and exhaustive. If events are not mutually exclusive, the probabilities would add up to a number greater than 1, and if they were not exhaustive, the sum of probabilities would be less than 1. Thus, there is a need to qualify this second property to ensure the events are properly defined (mutually exclusive, exhaustive). On an exam question, if the probabilities in a research study are added to a number besides 1, you might question whether this principle has been met.
These terms refer to the particular approach an analyst has used to define the events and make predictions on probabilities (i.e. the likelihood of each event occurring). How exactly does the analyst arrive at these probabilities? What exactly are the numbers based upon? The approach is empirical, subjective or a priori.

Empirical Probabilities
Empirical probabilities are objectively drawn from historical data. If we assembled a return distribution based on the past 20 years of data, and then used that same distribution to make forecasts, we have used an empirical approach. Of course, we know that past performance does not guarantee future results, so a purely empirical approach has its drawbacks.

Subjective Probabilities
Relationships must be stable for empirical probabilities to be accurate and for investments and the economy, relationships change. Thus, subjective probabilities are calculated; these draw upon experience and judgment to make forecasts or modify the probabilities indicated from a purely empirical approach. Of course, subjective probabilities are unique to the person making them and depend on his or her talents - the investment world is filled with people making incorrect subjective judgments.

A Priori Probabilities
A priori probabilities represent probabilities that are objective and based on deduction and reasoning about a particular case. For example, if we forecast that a company is 70% likely to win a bid on a contract (based on an either empirical or subjective approach), and we know this firm has just one business competitor, then we can also make an a priori forecast that there is a 30% probability that the bid will go to the competitor.
 Exam Tips and Tricks Know how to distinguish between the empirical, subjective and a priori probabilities listed above.

Stating the Probability of an Event as Odds "For" or "Against"

Given a probability P(E),

Odds "FOR" E = P(E)/[1 - P(E)] A probability of 20% would be "1 to 4".

Odds "AGAINST" E = [1 - P(E)]/P(E) A probability of 20% would be "4 to 1".
Example:
Take an example of two financial-services companies, whose publicly traded share prices reflect a certain probability that interest rates will fall. Both firms will receive an equal benefit from lower interest rates. However, an analyst's research reveals that Firm A's shares, at current prices, reflect a 75% likelihood that rates will fall, and Firm B's shares only suggest a 40% chance of lower rates. The analyst has discovered (in probability terms) mutually inconsistent probabilities. In other words, rates can't be (simultaneously) both 75% and 40% likely to fall. If the true probability of lower rates is 75% (i.e. the market has fairly priced this probability into Firm A's shares), then as investors we could profit by buying Firm B's undervalued shares. If the true probability is 40%, we could profit by short selling Firm A's overpriced shares. By taking both actions (in a classic pairs arbitrage trade), we would theoretically profit no matter the actual probability since one stock or the other eventually has to move. Many investment decisions are made based on an analyst's perception of mutually inconsistent probabilities.
Unconditional Probability
Unconditional probability is the straightforward answer to this question: what is the probability of this one event occurring? In probability notation, the unconditional probability of event A is P(A), which asks, what is the probability of event A? If we believe that a stock is 70% likely to return 15% in the next year, then P(A) = 0.7, which is that event's unconditional probability.

Conditional Probability
Conditional probability answers this question: what is the probability of this one event occurring, given that another event has already taken place? A conditional probability has the notation P(A | B), which represents the probability of event A, given B. If we believe that a stock is 70% likely to return 15% in the next year, as long as GDP growth is at least 3%, then we have made our prediction conditional on a second event (GDP growth). In other words, event A is the stock will rise 15% in the next year; event B is GDP growth is at least 3%; and our conditional probability is P(A | B) = 0.9.
Joint Probability

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