Probability after information is given

“Given that” narrows the possible outcomes. Restrict the sample space first, then divide the favourable count by the total within that restricted group. The denominator must reflect the information already known.

Example

Given that a selected student travels by bus, the probability they are senior is \(22/40=11/20\). Given that they are senior, the probability of bus travel is \(22/30=11/15\). These reverse questions have different denominators.

On a probability tree, later branches already describe the possibilities after earlier events. Without replacement, both the remaining counts and the total change. For a condition involving complete paths, add the relevant joint path probabilities and compare with all paths allowed by the condition.