Probability & Bayes Lab
Watch prior belief branch into competing likelihoods, keep only paths compatible with evidence, normalize them, and arrive at the posterior.
A deterministic probability-tree lab for conditional probability and Bayes' rule. A generic signal-detection example turns prior, sensitivity, false-positive rate, joint path probabilities, likelihood ratios, and posterior odds into one inspectable calculation.
Step by step
- Set the prior P(H), the probability of the hypothesis before observing the signal.
- Set P(+|H) and P(+|¬H) to describe how the same evidence can arise under two competing states.
- Observe either positive or negative evidence and keep only compatible tree branches.
- Multiply parent and branch probabilities to obtain joint path probabilities.
- Normalize those compatible paths to obtain P(H|evidence).
- Convert probabilities to odds and verify posterior odds = prior odds × likelihood ratio.
Core formulas
Bayes' theorem
P(H|E) = P(E|H)P(H) / P(E)Reverse the conditioning by weighting the prior with the evidence likelihood and normalizing.
Total evidence
P(E)=P(E|H)P(H)+P(E|¬H)P(¬H)The denominator counts every mutually exclusive way the observed evidence could occur.
Odds form
O(H|E)=O(H)·LR(E)Evidence multiplies prior odds by its likelihood ratio.
When to use Probability & Bayes Lab
- Learning conditional probability and the base-rate effect.
- Understanding why sensitivity or accuracy alone does not determine a posterior probability.
- Connecting probability trees, joint probabilities, normalization, odds, and likelihood ratios.