Conditional Probability practice problems

Recognition audit

Signals that this is the right method

1. The question says “given,” “among,” “of those,” or otherwise restricts attention to a known subgroup.
2. A probability must be updated after observing a test result, category, threshold, or earlier-stage outcome.
3. The data are supplied by rows or branches and the requested direction reverses the direction in which the rates were given.
4. Several latent cases can produce the observed event, so the denominator must combine those cases before Bayes’ theorem is applied.

Before calculation

A disciplined first pass

Name the requested quantity, mark its units or time point, and identify the assumption that permits the formula. Only then insert the numerical inputs. This separates model errors from arithmetic errors and makes the answer-choice comparison auditable.

Conditional probability

Both numerator and denominator are measured inside the same original probability model.

Law of total probability

Use a genuine partition: the cases are disjoint and together exhaust the sample space.

Bayes’ theorem

The denominator is the total probability of the observed evidence, not merely the most likely branch.