A tree of results
Two coins are tossed, one after the other.
- The trunk splits into heads and tails, each 1/2.
- Each branch splits again into heads and tails, each 1/2.
- Multiply along a path: heads then heads = 1/2 × 1/2 = 1/4.
- Four paths, each 1/4. They add to exactly 1.
One stage per toss
Tree diagram — a drawing of an experiment with several steps. Stage — one step, here one toss. The trunk is the start. A branch is one result of a stage. The first toss splits the trunk into two branches. The second toss splits each branch into two more.
Path — a full route from trunk to end. There are four: HH, HT, TH, TT. One for each pair of results.
Multiply along a path
A path's chance is found by multiplying along it. Heads then heads: 1/2 × 1/2 = 1/4. For a fair coin, heads is 1/2 of the tosses. Then heads again is 1/2 of those. The second toss is not changed by the first. Every path here gives 1/4. The four paths add to 4 × 1/4 = 1.
"Exactly one head" is the paths HT and TH. 1/4 + 1/4 = 1/2.
Rule: One stage per choice. Multiply along a path. The paths together add to exactly 1.
Branches need not be equal. Bag X has 2 red beads and 1 blue bead. Bag Y has 2 white beads and 1 black bead. Pick one bead from each bag without looking. In bag X, red is 2/3 and blue is 1/3. In bag Y, white is 2/3 and black is 1/3. The four paths are:
- Red then white: 2/3 × 2/3 = 4/9.
- Red then black: 2/3 × 1/3 = 2/9.
- Blue then white: 1/3 × 2/3 = 2/9.
- Blue then black: 1/3 × 1/3 = 1/9.
They add to 4/9 + 2/9 + 2/9 + 1/9 = 1.
Pick a branch at each stage. Find all four paths.
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