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

Bayes’ formula is used to calculate an updated or posterior probability given a set of prior probabilities for a given event. It’s a theorem named after Reverend T Bayes and it is widely used in Bayesian methods of statistical inference.

This is the logic used to come up with the formula:

Let E1, E2, E3, …, En be a set of mutually exclusive and exhaustive events.

Using the conditional probability:

$$P(E_i | A) =\cfrac {P(E_i A)}{P(A)}$$

And also the relationship:

$$P(E_iA) = P(AE_i) = P(E_i)P(A | E_i)$$

And the total probability rule:

$$P(A) = \sum {P(AE_j)} \quad \text { for all j} = 1, 2,…,n$$

We can finally substitute for P(EiA) and P(A) in equation 1

This gives:

$$P(E_{ i }|A)=\cfrac { P(E_{ i })P(A|E_{ i }) }{ \sum _{ j=1 }^{ n }{ P(E_{ i })P(A|E_{ i }) } }$$

This is the Bayes’ formula and it allows us to ‘turnaround’ conditional probabilities i.e., we can calculate P(Ei | A) if given information only about P(A | Ei).

Note:

1. P(Ej) are known as prior probabilities;
2. the event A is some event known to have occurred; and
3. P(Ei | A) is the posterior probability.

## Example: Bayes’ Formula

A Civil Engineer wishes to investigate the punctuality of electric trains by considering the number of train journeys. In the sample, 50% of trains were destined for New York, 30% Vegas and 20% Washington DC. The probabilities of a train arriving late in New York, Vegas, and Washington DC are 40%, 35%, and 25% respectively. If the Engineer picks a train at random from this group, what is the probability that the destination of the train would be New York?

Solution:

We are looking for P(New York | Late).

Let’s define the following events:

First, N is the event “A train chosen at random whose destination is New York.”

Secondly, V is the event “A train chosen at random whose destination is Vegas.”

And W is the event “A train chosen at random whose destination is Washington DC.”

Finally, let L be the event “A randomly chosen train arrives late.”

\begin{align*} P(N|L) & =\cfrac { P(N)P(L|N) }{ P(N)P(L|N)+P(V)P(L|V)+P(W)P(L|W) } \\ & =\cfrac { 0.5\ast 0.4 }{ 0.5\ast 0.4+0.3\ast 0.35+0.2\ast 0.25 } \\ & =\cfrac { 0.2 }{ 0.355 } \\ & =0.5634 \\ & =56.3\% \\ \end{align*}

We have computed P(N | L) given only P(L | N), hence the phrase ‘turnaround conditional probability’.

## Question

A chartered analyst can choose any one of these three routes, A, B or C, to get to work. The probabilities that she arrives on time using routes A, B, and C are 50%, 52%, and 60% respectively. If she is equally likely to choose any one of the routes and arrive on time, calculate the probability that she chose route A.

A. 30.9%

B. 16.67%

C. 25%

Solution

First, you should define the relevant events:

let: A be the event, “Chooses route A”;

B the event, “Chooses route B”; and

C the event, “Chooses route C”.

Lastly, define event T as, “Arrives to work on time”.

Now, what we have is P(T | A) =  i.e. the probability that the analyst arrives on time given that she chooses route A.

However, we want to find the turnaround probability P(A | T) i.e. the probability that the analyst chooses route A  given that she arrives on time.

Hence, applying Bayes’ formula:

\begin{align*} P(A|T) & =\cfrac { P(A)P(T|A) }{ P(A)P(T|A)+P(B)P(T|B)+P(C)P(T|C) } \\ & =\cfrac { \frac {1}{3} \ast 0.5 }{ \frac {1}{3} \ast 0.5+\frac {1}{3} \ast 0.52+\frac {1}{3} \ast 0.6 } \\ & =\cfrac { 0.16667 }{ 0.54 } \\ & =0.30865 \\ & =30.9\% \\ \end{align*}

Tip: good understanding of Bayes’ theorem can only be preceded by a good understanding of the rules of probability. The focus should be on the calculations.

Calculate and interpret an updated probability using Bayes’ formula.

Shop CFA® Exam Prep

Offered by AnalystPrep Level I
Level II
Level III
All Three Levels
Featured Shop FRM® Exam Prep FRM Part I
FRM Part II
FRM Part I & Part II
Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Shop Actuarial Exams Prep Exam P (Probability)
Exam FM (Financial Mathematics)
Exams P & FM
Shop GMAT® Exam Prep Complete Course 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.