Tree diagrams, conditional probability and Venn probability

Last time

Before we start: last time, in Single events and relative frequency, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Two draws, four routes

You take one counter from a bag, keep it, and take another. Each draw can go two ways, so there are four routes in all. A tree diagram lays them out. The rule is simple: multiply along a route, add between routes. And when the first draw changes the second, we call it conditional probability.

Building a tree

Draw one branch for each outcome. Write its probability on the branch. The branches from one point add to one. The second set of branches grows from the end of each first branch. Write the outcomes at the end of each route, ready to multiply.

With replacement

A bag has three red and two blue counters. You take one, replace it, and take another. The probabilities stay three fifths and two fifths. The chance of two reds is three fifths times three fifths: nine twenty-fifths. The chance of at least one blue is one minus the chance of two reds: sixteen twenty-fifths.