Equations · Probability and statistics
Law of total probability
The probability of an event, built up from the probabilities of the separate cases that lead to it.
P(A) = ∑ᵢ P(A | Bᵢ) P(Bᵢ)
Aethel says
P of A equals the sum over i of P of A given B sub i, times P of B sub i
What it says
Split the possibilities into cases B₁, B₂ and so on that cannot happen together and that cover everything. The chance of A is the chance of A in each case, weighted by how likely that case is, added up.
It supplies the denominator of Bayes' theorem.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| ∑ᵢ | the sum over i of | add up the terms for every case i |
| Bᵢ | B sub i | the ith case; together the cases cover all outcomes, without overlap |
| P(A | Bᵢ) | P of A given B sub i | the probability of A in case i |
More on∑Summation (sigma notation)|Vertical bar: given, divides
Where you meet it
Probability courses, risk models and Bayesian statistics.
Questions
How do you say Law of total probability out loud?
P(A) = ∑ᵢ P(A | Bᵢ) P(Bᵢ) is read: “P of A equals the sum over i of P of A given B sub i, times P of B sub i.” That is exactly what Aethel says when it meets the formula in a book.
What does ∑ᵢ mean in Law of total probability?
∑ᵢ (said “the sum over i of”) is add up the terms for every case i.
What does Bᵢ mean in Law of total probability?
Bᵢ (said “B sub i”) is the ith case; together the cases cover all outcomes, without overlap.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.