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

# Conditional Probability

## Unconditional Probability

Unconditional probability (also known as marginal probability) is simply the probability that the occurrence of an event does not, in any way, depend on any other preceding events. In other words, unconditional probabilities are not conditioned on the occurrence of any other events; they are ‘stand-alone’ events. Therefore, if we are interested in the probability of an event, say, A, the standard annotation is P(A). Let’s look at a few examples:

1. The probability that it will rain on a given day without considering the rainfall pattern of a given area or any other climatic factor.
2. The probability that a given stock will earn a 10% annual return without considering the preceding annual returns.

We will learn in a future learning outcome statement to calculate and interpret an unconditional probability using the total probability rule.

## Conditional Probability

Conditional probability is the probability of one event occurring with some relationship to one or more other events. Our interest lies in the probability of an event ‘A’ given that another event ‘B ‘has already occurred. Here’s what you should ask yourself:

“What is the probability of one event occurring if another event has already taken place?”

We pronounce P(A | B) as “the probability of A given B.,” and it is given by:

$$P(A│B)=\frac{P(A\cap B)}{P(B)}$$

The bar sandwiched between A and B simply indicates “given.”

#### Example: Groups of Investors

In a group of 100 investors,

• 30 purchase bonds; and
• 20 purchase stocks and bonds.

If an investor chosen at random bought bonds, what is the probability they also bought stocks?

Solution

$$\begin{array}{c|c|c} \textbf{Event} & \textbf{Notation} & \textbf{Probability} \\ \hline \text{Buys stocks} & \text{A} & \text{0.4(=40/100)} \\ \text{Buys bonds} & \text{B} & \text{0.3 (=30/100)} \\ \text{Buys stocks and bonds} & \text{A and B} & \text{0.2 (=20/100)} \\ \end{array}$$

We want the probability of an investor buying stocks given that they have already bought bonds. This is P(A | B):

\begin{align} P\left( A|B \right) & =\frac { P\left( A\cap B \right) }{ P\left( B \right) } \\ & =\frac { 0.2 }{ 0.3 } = 0.67 \end{align}

### Rearranging the Formula

Note that we can also make the numerator the subject so that:

$$P\left( A\cap B \right) =P\left( A|B \right) P\left( B \right)$$

For independent events, which we will see later, however,

$$P\left( A|B \right) =\frac{P(A)P(B)}{P(B)}=P\left( A \right)$$

This is also true for $$P(B│A)=P(B)$$.

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
2021-07-23
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.
diana
2021-07-17
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
2021-07-16
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
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected
Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
Daniel Glyn
2021-03-24
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
2021-03-18
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.