Limited Time Offer: Save 10% on all 2022 Premium Study Packages with promo code: BLOG10

Define and calculate conditional probabilities

Define and calculate conditional probabilities

Conditional Probability

Given two events \(A\) and \(B\), the conditional probability of event \(A\) occurring, given that event \(B\) has occurred, is the probability of event \(A\) and \(B\) occurring over the probability of event \(B\) occurring as shown by the formula below:

$$ P\left( A|B \right) =\frac { P\left( A\cap B \right) }{ P\left( B \right) } $$
as long as \(P (B) > 0\).

As shown in the diagram below, the probability of \(A\) given \(B\) occurred is the shaded region \(A \cap B\).

Example: Conditional ProbabilityExample: Conditional Probability

Suppose 50% of the students in a school take geometry, 30% take history, and 15% take both geometry and history. Given that a student selected at random takes geometry, what is the probability he or she also takes history?


Let event \(A\) be “takes geometry” and event \(B\) be “takes history.”

\(P \left(B \right) = 0.30\)

\(P \left(A \cap B \right) = 0.15\)

$$ P\left( B|A \right) =\frac { P\left( A\cap B \right) }{ P\left( A \right) } ={ 0.15 }/{ 0.50 }=0.30\quad or\quad 30\% $$

This concept can be extended to more than 2 events as shown in the formula for 3 events below:

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

Example: Conditional Probability

Suppose that the students at a university are grouped such that:

 45% take chemistry

 35% take biology

 25% take math

 10% take chemistry and math

 5% take chemistry and biology

 3% take biology and math

 1% take all three subjects

Calculate the probability that a student takes math, given that they take chemistry and biology.

$$P\left( \text{Math|Chemistry and Biology} \right) =\frac{{ P\left( \text{Math and Chemistry and Biology} \right) }}{{ P\left( \text{Chemistry} \right) \times P\left( \text{Biology|Chemistry} \right) }}$$

$$\begin{align*} P\left( \text{Biology|Chemistry} \right) & =\frac{{ P\left( \text{Biology and Chemistry} \right) }}{{ P\left( \text{Chemistry} \right) }} \\ & ={ .05 }/{ .45 }={ 1 }/{ 9 } \\ \end{align*} $$

$$P(\text{Math|Chemistry and Biology})=\frac{{ .01 }}{{ \left( { .45\times .05 }/{ .45 } \right) }}={ .01 }/{ .05 }={ 1 }/{ 5 }=.20\quad or\quad 20\%$$

The definition of conditional probability can be rewritten to come up with multiplication rule for probabilities:

$$P(A\cap B)=P(A|B)\bullet P(B)$$

Also, if the events A and B are independent, then:


Learning Outcome

Topic 1.e: General Probability – Define and calculate conditional probabilities.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop GMAT® Exam Prep

    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.
    Nyka Smith
    Nyka Smith
    Every concept is very well explained by Nilay Arun. kudos to you man!
    Badr Moubile
    Badr Moubile
    Very helpfull!
    Agustin Olcese
    Agustin Olcese
    Excellent explantions, very clear!
    Jaak Jay
    Jaak Jay
    Awesome content, kudos to Prof.James Frojan
    sindhushree reddy
    sindhushree reddy
    Crisp and short ppt of Frm chapters and great explanation with examples.