Tree diagrams

A tree diagram shows every outcome of a multi-step experiment, like tossing a coin twice:

• from a starting point, draw a branch for each outcome of step 1;
• from the end of each, draw a branch for each outcome of step 2;
• and so on.

Each path from the start to an end is one complete outcome. Two tosses give 4 paths: HH, HT, TH, TT.

Multiply along a path. Write each branch's probability on it. For a fair coin each branch is 1/2, so each path is 1/2 × 1/2 = 1/4.

Add across the paths you want. One head and one tail is HT + TH = 1/4 + 1/4 = 1/2.

Branches need not be equal. A pen box of 3 red, 4 black and 2 green gives branches 3/9, 4/9, 2/9.

With or without putting it back? If the first pen is put back, the second pick has the same branches. If it is not put back, one pen has gone, so the total drops to 8 and the branches change.