Conditional probability and Bayes' theorem

Parametric equations

$P(A\mid B)$ is how likely $A$ is once $B$ has already happened. Bayes’ theorem ties that probability to $P(B\mid A)$, the probability of the result given the cause. It is the core of how diagnosis, screening, and machine learning update a belief.

Conditional probability:

Bayes’ theorem:

The law of total probability is often written

and that sum is how the denominator expands.

Interaction

• Prior P(A): change the probability before the evidence, and watch the same evidence lead to a different posterior
• Conditional P(B|A) and Conditional P(B|¬A): change the two conditional probabilities, and compare how a false positive shifts the conclusion
• Tree, Area model, and Bayes update: switch among branch products, area ratios, and the bars for $P(A)$, $P(B)$, and $P(A\mid B)$
• Medical test, Card draw, and Spam: replace the letters $A$ and $B$ with a concrete pair of events

Related

• Binomial distribution to the normal distribution
• Buffon’s needle

Further reading

• Bayes’ theorem (Wikipedia)