Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

Introduction to Probability Concepts

Introduction to Probability Concepts

Probability is a measure of the likelihood that something will happen. We usually express probabilities as percentages, from 0% (impossible to happen) to 100% (guaranteed). In fact, we can express almost any event as a probability. For instance, we can gauge the likelihood that it will rain on a given day or the likelihood of one passing one’s examination.

Estimation of the probabilities of anticipated outcomes is critical in financial management. More precisely, concepts in probability help analysts quantify risks. The following are the major probability concepts you should know.

Random Variable

A random variable is any quantity whose expected future value is not known in advance. Examples include the expected future value of a mutual fund, expected dividend payment on a stock, or the expected standard deviation of investment returns. We cannot know the exact future value of an investment since it is usually a function of multiple factors, some of which may be beyond the control of the analyst.

Note that returns that are fixed cannot be described as random variables. For instance, if a risk-free government bond is quoted at a fixed discount of, say, 6%, the bond’s future value can be calculated in advance. As such, it is not a random variable.


An outcome is any possible value that a random variable can take. So, for example, if you roll a six-sided die, there are six possible outcomes since any number, from 1 to 6, can come up.

If a ‘3’ comes up, then that is an outcome. In case a ‘5’ comes up, it is another outcome. So, for example, if a stock offers shareholders a $2 dividend per share at the end of a year, then the $2 dividend is an outcome.


A single outcome or a set of outcomes is known as an event. If we take the die example above, rolling a 4 is an event, rolling an odd number is another event, etc.

Mutually Exclusive Events

Mutually exclusive events are such that one event precludes the occurrence of all the other events. For example, if you roll a die and a 4 comes up, that particular event precludes all the other events, i.e., { 1, 2, 3, 5, and 6 }. In other words, rolling a 1 and a 5 are mutually exclusive events: they cannot occur simultaneously.

Furthermore, there is no way a single investment can have more than one arithmetic mean return. As such, an arithmetic return of, say, 20% constitutes a mutually exclusive event.

Exhaustive Events

Events are said to be exhaustive if they include all possible outcomes. Suppose we roll a die once and categorize the outcomes into two events as follows:

{1, 3, 5}   {2, 4, 6}

Each of the above sets of outcomes is an event. The two events are exhaustive because they include all the possible outcomes.

Example: Random Variable, Outcome, and Event

We do a random experiment of drawing one card from a deck. It consists of a sample space of 52 elements (i.e., cards).

A pack or deck of playing cards has 52 cards which are divided into four categories as Spades (♠) Clubs (♣) Hearts (♥), and Diamonds (♦). Each of these categories has 13 cards, 9 cards numbered from 2 to 10, an Ace, a King, a Queen, and a Jack. Hearts and Diamonds are red-faced cards whereas Spades and Clubs are black-faced cards. Kings, Queens, and Jacks are called face cards.

  • A random variable, in this case, will result from drawing a card from a deck of shuffled cards since there is uncertainty in regard to the card that might be drawn.
  • Suppose a card is drawn randomly from a deck and turns out to be an Ace, then, in this case, outcome is “Ace.”
  • Drawing two spades and five hearts from the deck qualifies to be an event.
Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Online Tutoring
    Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

    Video Lessons

    Sergio Torrico
    Sergio Torrico
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.