Conditional Probability, Total Probability and Bayes' Theorem

IB AA: conditional probability and Bayes

Conditional probability and finite tree-based total probability are core at SL and HL. Bayes’ theorem is additional HL content (AHL4.13), with problems involving at most three source events. The general partition formula below is supporting notation.

Conditional probability

$P(A\mid B)$ is the probability that $A$ occurs given that $B$ has occurred. The sample space is restricted to $B$, and $P(B)\ne0$.

Conditional and intersection forms

For a finite sample space with equally likely outcomes, the count ratio gives $P(A\mid B)=\dfrac{n(A\cap B)}{n(B)}$. In general, for $P(B)>0$,\[P(A\mid B)=\dfrac{P(A\cap B)}{P(B)},\qquad P(A\cap B)=P(A\mid B)P(B).\]

Law of total probability

Let $B_1,B_2,\ldots,B_n$ be mutually exclusive and exhaustive, so $P(B_i\cap B_k)=0$ for $i\ne k$ and $\sum_{i=1}^{n}P(B_i)=1$. Then, for any event $A$, Assume $P(B_i)>0$ for every conditioning event; omit any zero-probability partition cells.

Total probability

\[P(A)=\sum_{i=1}^{n}P(A\cap B_i)=\sum_{i=1}^{n}P(A\mid B_i)P(B_i)\]

Bayes' theorem

Bayes' rule

For $P(A)>0$ and $P(B)>0$, \[P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}\]

Here $P(A\mid B)$ is the posterior probability, $P(A)$ is the prior probability, $P(B)$ is the total probability, and $P(B\mid A)$ is the likelihood.