Glossary: M04 — Probability Concepts
Module: M04 Formulas: Formula Sheet Concept page: Probability
Random Variable
A variable whose value is determined by the outcome of a random process (an experiment). Can be discrete or continuous.
LOS: 4.a | Examples: The return on a stock next year; the number of defaults in a bond portfolio.
Outcome
A possible value of a random variable or a possible result of a single trial of an experiment. Outcomes are the basic building blocks of probability.
LOS: 4.a | Related: Event, Sample Space
Event
A specified set of one or more outcomes of a random experiment. Events can be simple (single outcome) or compound (multiple outcomes).
LOS: 4.a | See: Mutually Exclusive Events, Exhaustive Events
Probability
A number between 0 and 1 that expresses the likelihood of an event occurring. A probability of 0 means the event is impossible; a probability of 1 means it is certain.
LOS: 4.a | See: Subjective Probability, Empirical Probability, A Priori Probability
Subjective Probability
A probability based on personal judgment or informed opinion rather than formal analysis or historical data. Not repeatable.
LOS: 4.a | Example: An analyst’s estimate that a merger will succeed has a 70% probability.
Empirical Probability
A probability estimated from observed historical data — the relative frequency of the event over a large number of trials.
LOS: 4.a | Example: If a stock rose on 120 of 200 trading days, the empirical probability of a rise is 120/200 = 60%.
A Priori Probability
A probability derived from logical analysis of equally likely outcomes, without reference to observed data. Based on deductive reasoning.
LOS: 4.a | Example: The probability of rolling a 4 on a fair die is 1/6 by a priori reasoning.
Mutually Exclusive Events
Events that cannot occur simultaneously. The occurrence of one event precludes the occurrence of the other.
LOS: 4.b | Related: Addition Rule
Exhaustive Events
A set of events that includes all possible outcomes. The sum of probabilities of exhaustive, mutually exclusive events equals 1.
LOS: 4.b
Odds For
The ratio of the probability that an event occurs to the probability that it does not occur.
LOS: 4.b | Example: If , odds for = 3 to 1 (or 3:1).
Odds Against
The ratio of the probability that an event does not occur to the probability that it does occur. The inverse of odds for.
LOS: 4.b | Example: If odds against are 4 to 1, then .
Unconditional Probability
The probability of an event without conditioning on any other event. Also called the marginal probability. Denoted .
LOS: 4.c | Contrast: Conditional Probability
Conditional Probability
The probability of event given that event has already occurred. Updated probability in light of new information.
LOS: 4.c | Foundation for: Bayes’ Formula, Total Probability Rule
Joint Probability
The probability that both events and occur simultaneously. Denoted or .
LOS: 4.c | Related: Multiplication Rule, Probability Tree
Addition Rule
The probability that event or event (or both) occurs.
For Mutually Exclusive Events:
LOS: 4.d | Related: Unconditional Probability
Multiplication Rule
The probability that both events and occur, derived from conditional probability.
For Independent Events:
LOS: 4.d | Related: Joint Probability
Independent Events
Two events are independent if the occurrence of one does not affect the probability of the other. and .
LOS: 4.e | Contrast: Dependent Events
Dependent Events
Events where the occurrence of one affects the probability of the other. .
LOS: 4.e | Example: The default of one firm in an industry may increase the probability of default by a competitor (contagion).
Total Probability Rule
Expresses the unconditional probability of an event as a weighted average of conditional probabilities, where the weights are the probabilities of the conditioning events.
where are mutually exclusive and exhaustive scenarios.
LOS: 4.f | Used in: Bayes’ Formula | Application: Scenario analysis in investment management.
Expected Value
The probability-weighted average of all possible outcomes of a random variable. The mean of the probability distribution.
LOS: 4.g | Key: Expected value does not have to be a possible outcome (e.g., expected number of children = 2.1). | Related: Variance of Random Variable
Variance of Random Variable
The expected value of the squared deviations from the mean. Measures the dispersion of a random variable’s probability distribution.
LOS: 4.g | Related: Standard Deviation of Random Variable, Sample Variance
Standard Deviation of Random Variable
The positive square root of the variance of a random variable. Expressed in the same units as .
LOS: 4.g | Related: Portfolio Standard Deviation
Bayes’ Formula
A method for updating the probability of an event given new information. Calculates the posterior probability from the prior probability and the likelihood of the new evidence.
Expanded using Total Probability Rule:
LOS: 4.h | Key: = prior probability; = posterior probability (updated after observing ).
Factorial
The product of all positive integers from 1 to . Used in counting methods. Notation:
LOS: 4.i | Related: Permutation, Combination
Permutation
The number of ways to arrange objects chosen from distinct objects, where order matters.
LOS: 4.i | Contrast: Combination — order does not matter.
Combination
The number of ways to choose objects from distinct objects, where order does not matter. Also called “n choose r.”
LOS: 4.i | Application: Number of ways to select a portfolio of stocks from candidates. | Related: Binomial Distribution
Probability Tree
A diagram that maps out all possible outcomes of a sequence of events, with each branch labeled with its probability. Visually represents joint and conditional probabilities.
LOS: 4.f | Use: Illustrates the Total Probability Rule and Bayes’ Formula calculations. | Related: Joint Probability, Conditional Probability