Conditional Probability and Tree Diagrams

Conditional Probability and Tree Diagrams

A tree diagram displays sequential outcomes. Probabilities on branches from one node add to 1; multiply along a path for an ‘and’ outcome, then add separate relevant paths for an ‘or’ event. Conditional probability means the second branch is based on what has already happened, so write the updated denominator or condition clearly.

Worked reasoning

A bag has 2 red and 3 blue counters. P(red then blue without replacement) is $2/5\times3/4=3/10$. The path uses the updated second branch after red is removed.

Exam method

1. Draw first branches that sum to 1. 2. Update second branches for conditions. 3. Multiply along desired paths, then add mutually exclusive paths when necessary.

Audit a completed tree

Before calculating, check that branches from each node sum to 1. After calculating, check that all terminal path probabilities sum to 1 too; this catches a missing outcome or copied denominator. Circle the event wording, then shade exactly the successful paths. For a conditional question such as $P(A\mid B)$, restrict attention to B first and renormalise within it. A correct-looking product can still answer the wrong event if this conditioning step is missed.

One-minute retrieval: Conditional Probability and Tree Diagrams

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.