Tree Diagrams and Combined Events

1 · Frequency trees: counting first

Before we use probabilities, let's use counts. A frequency tree starts with a whole group, splits it into two parts, then splits each part again. Each box holds a frequency (how many). The one rule: the two boxes coming out of a box add up to that box. So whenever a split has one gap, you can fill it by subtracting.

Fill in the four empty boxes from the information given, then press Check. The tree updates as you type.

Once the tree is full, a probability is just a box divided by the box it came from at the start: "a person is picked at random from everyone" means divide by the very first number. Frequency trees are the bridge to probability trees: replace each count by a fraction of the box before it and you get the branches of section 3.

2 · Independent events

A combined event is two (or more) things happening together, like "spin red and roll a 6". Two events are independent if one happening does not change the probability of the other. A spinner can't see what the dice did, so they're independent. Events that do affect each other are dependent.

For independent events there's a short-cut: P(A and B) = P(A) × P(B). To see why it's a multiplication, draw every equally likely outcome as a grid: spinner sections across, dice faces down. Every cell is equally likely. The orange cells are "red and low enough": a block of (red columns) × (good rows) cells.

So the multiplication isn't a rule to memorise: it's counting cells in a rectangle. It only works because the spinner doesn't change the dice. Spotting whether events are independent is a skill in itself, and exam questions sometimes ask what you have assumed. Decide for each pair:

3 · Tree diagrams: the AND rule and the OR rule

A tree diagram shows a combined event in stages. From the start, draw one branch for each outcome of the first stage and write its probability on the branch. From the end of each branch, draw the branches for the second stage. Each route from the start to an end is a path, and each path is one combined outcome.

Two rules do everything:

• AND: multiply along the branches. To go down a path, the first thing and the second thing must happen, so multiply (that's the grid in section 2).
• OR: add the end results. Different paths can't happen at the same time (they're mutually exclusive), so if several paths give what you want, add their end probabilities.

Here a biased spinner lands on red (R) or not red (R′), and a biased coin lands heads (H) or tails (T). Set the probabilities, then tap the end results you want (or use a button) and watch the working.

4 · Building a tree from a bag

A bag holds red and blue counters. You take one at random, note its colour, then take a second. There are two ways to do it:

• With replacement: put the first counter back before the second pick. The bag is the same both times, so the picks are independent.
• Without replacement: keep the first counter out. Now the bag has one fewer counter for the second pick, and one fewer of whatever colour you took. The second pick depends on the first. (Higher: the second-stage branches are conditional probabilities; see conditional probability.)

Choose the bag, then press Next step to build the tree one stage at a time. Switch between with and without replacement and watch the second-stage branches change.

5 · Dependent events: the denominator drops

Here is "without replacement" happening one counter at a time. Take counters out and watch the fraction for the next pick. The bottom number always goes down by 1 (one fewer counter). The top number goes down by 1 only if you took that colour. Under the bag, the probability of the exact sequence you took is built up by multiplying.

6 · "At least one" = 1 − P(none)

"At least one six" means one six, or two sixes, or more. On a tree that's almost every path, which is a lot of adding. But there is only one path with no six at all. Since all the paths add up to 1:

P(at least one) = 1 − P(none)

Roll a fair dice more times and watch the tree. S means a six (probability 1/6) and N means not a six (5/6). Orange paths have at least one six; only the grey one doesn't.

7 · Three-stage trees

Nothing new, just a bigger tree. Each path now has three branches to multiply, and with two outcomes at each stage there are 2 × 2 × 2 = 8 paths. Questions like "exactly two" are where trees earn their keep: you have to find every path with exactly two, and there are three of them. Pick an experiment, then tap paths or use a button.

8 · Spot the mistake

Each student below has made one of the classic tree-diagram mistakes. Look at their tree and working, and pick what went wrong.

Worked examples

Quick check

What you've learned

• A frequency tree splits a group twice; the two boxes from any box add up to it. A probability is a box over the starting total.
• Events are independent if one doesn't change the probability of the other. Then P(A and B) = P(A) × P(B), which is counting cells in a grid.
• On a tree diagram, multiply along a path (AND) and add the end results of different paths (OR).
• Branches from one point add to 1, and all the end results add to 1. Use this to fill gaps and to check.
• Without replacement makes the events dependent: on the second stage the bottom drops by 1, and the top drops for the colour already taken.
• At least one = 1 − P(none).
• Three-stage trees work the same way: multiply three branches; for "exactly two", find every path (often three) and add.