See conditional probability. Check every path. Trust the posterior.

Bayes Tree Lab is an iPhone app that helps you learn and verify two-stage conditional probability problems. Draw the branch tree, expand Bayes' theorem line by line, and test the theory with repeatable simulations.

What you can do

1. Edit a probability tree

Set the prior P(A) and the two conditional probabilities P(B | A) and P(B | not A) using a slider or the keyboard. Sibling branches rebalance automatically, so each pair of branches always sums to one.

2. Audit every path

Each of the four paths shows its joint probability with the multiplication written out. A built-in check confirms the paths add up to one.

3. Expand the Bayes posterior

Pick the outcome you observed, then follow three steps:

1. Joint probabilities
2. Evidence probability
3. Posterior probability

If an observation is impossible, the app tells you why the posterior is undefined.

4. Look at it as odds

Prior odds times the likelihood ratio gives the posterior odds. A log-odds axis shows how far the evidence moved your belief, and the result is cross-checked against the standard posterior.

5. Think in natural frequencies

See the same problem as head-counts in a population of 100, 1,000, or 10,000 people. Counting is often easier to reason about than fractions.

6. Simulate with a seed

Choose a random seed and up to one million trials. The same seed and inputs always give the same result. The app compares expected and observed counts for every path and runs a chi-square goodness-of-fit test.