Supplied branches and model decisions

Use the probability for the branch you are on

A later success probability may be supplied separately after different first-stage outcomes. These values belong to different branches; do not average them without accounting for how likely each branch is.

If a first stage produces \(A\) or \(B\), the second-stage outcome \(S\) can be reached by \(AS\) or \(BS\). Find each complete path probability, then add them. These paths are disjoint because the first-stage outcomes differ.

A probability tree can model dependent or independent trials. It is the branch probabilities and the selection rules, not the tree’s shape, that determine dependence.

For a decision, calculate the probability under each proposed model and compare it with the stated limit. Explain which assumptions must remain valid. Separate events are not automatically independent when a shared cause can affect them both.

For a fixed three-stage experiment, a phrase such as “the first success is on the third trial” describes failure, failure, success. “At least one success” is often found more efficiently from its complement, no successes.