Conditional probability and Bayes' rule

Updating on information

The conditional probability of $A$ given $B$ is $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$: restrict attention to the cases where $B$ happened and ask how often $A$ also happened. Rearranged it is the multiplication rule, $P(A\cap B)=P(A\mid B)\,P(B)$. Events are independent exactly when $P(A\mid B)=P(A)$.

The law of total probability splits $B$ over a set of exclusive, exhaustive cases $A_i$. Bayes' rule then reverses a conditional: from how likely the evidence is in each case, $P(B\mid A)$, it gives how likely each case is given the evidence, $P(A\mid B)$. The starting probability $P(A)$ is the prior, and the result is the posterior. The most common mistake is to ignore the prior (the base rate).

• Conditional: $P(A\mid B)=P(A\cap B)/P(B)$, the probability inside the world where $B$ happened.
• Total probability: add up $P(B\mid A_i)P(A_i)$ over every case.
• Bayes: posterior = likelihood × prior ÷ evidence. Do not confuse $P(B\mid A)$ with $P(A\mid B)$.

Lab · what's graded

• A conditional probability15 pts
• The chance of a flag15 pts
• The posterior20 pts
• The transposed conditional15 pts
• The steps of a Bayes update15 pts
• A 90% test that is mostly wrong20 pts