GCSE · Maths · AQA · Spec 8300 · Foundation

Tree diagrams (Additional Foundation)

Grab two marbles from a bag at the same time. Is that one event or two? Here's the trick that makes it easy: pretend one came out first.

Maths · Probability

Two marbles from one bag

Start with the marble going back in the bag. Then switch to 'Without replacement' and watch the second-pick fractions change. Tap any row to see its path light up.

A bag holds 4 red and 2 blue marbles. You take two marbles out.

1st draw2nd draw4/62/64/62/64/62/6RBRBRBRRRBBRBB4R · 2B4R2B

The first marble goes back, so the bag is 4 red and 2 blue again. The second pick has exactly the same fractions as the first: the picks are independent.

Tap a row

Tap an outcome row — the two branches on its path glow indigo and the multiplication is laid out.

Watch out: Two marbles taken out 'at once' is the same as taking one and then another straight away without putting it back. Use the 'Without replacement' tree.

Maths · Listing outcomes

Every end is one outcome

Walk every path: the coin first, then the spinner. Work top to bottom so nothing gets missed.

Coin → Spinner

2 × 3 = 6 possible outcomes, and the tree ends 6 times.

On Start. 2 branches to choose from.

Strip the probabilities off and a tree is simply a list. Flip a coin, then spin a spinner numbered 1, 2 and 3. Walk each path and collect what's at its end.

Watch out: A two-way grid (coin down the side, spinner across the top) lists the same 6 outcomes as 6 cells, and that's fine for two stages when every outcome is equally likely. The tree wins when there's a third stage, or when the branches need different probabilities on them.

Maths · Add or multiply?

One path, or several?

A bag has red and blue marbles and you take two. Which kind of question is each one?

Still to sort

One path: multiply along it (0)

The question describes one exact sequence — this, then that.

Where the line is: If the order is fixed, there is only one path, however the question is worded.

Several paths: multiply each, then add (0)

More than one end of the tree gives what you want.

Where the line is: Words like 'either order', 'one of each', 'the same colour' or 'at least one' usually hide more than one path.

6 of 6 still to sort.

The arithmetic is easy. The skill is reading the question and knowing how many paths it wants. Sort each one before you calculate anything.

Maths · Two at once

What if they come out together?

A box holds 6 milk chocolates and 3 dark chocolates. Sam grabs two at the same time without looking.

How would you find the probability that both are dark? Pick the idea closest to yours.
How sure are you?

Maths · Reading a tree

Can't both happen? Or doesn't affect the other?

For each pair of events, tick 'Mutually exclusive' if they can't both happen, and 'Independent' if one happening doesn't change the chance of the other.

Getting red-blue, and getting blue-red, on the same two picks
Red on the first pick, and blue on the first pick
Red on the first pick, and red on the second pick — the marble is put back
Red on the first pick, and red on the second pick — the marble is not put back
Heads on a coin, and a 3 on a spinner

Maths · Your turn

Finish the method

A drawer holds 4 black socks and 3 white socks. Two socks are taken out at random. Find the probability that they are different colours.

  1. Two socks taken out = one sock, then a second sock from the 6 that are left.
  2. First sock: P(black) = 4/7 and P(white) = 3/7.
  3. missing step
Which line is step 3?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two picks, one picture: multiply along a path, add the ends you want.

What you need to know

  • A tree diagram lists every outcome of a combined experiment: each end of the tree is one outcome.
  • Branches from the same point add up to 1.
  • Multiply along the branches of a path to find the probability of that outcome.
  • Add the probabilities of the ends when more than one path gives the result you want.
  • With replacement, the second-stage branches are the same as the first (independent events). Without replacement, both the count and the total drop by one (dependent events).
  • Taking two items at once is the same as taking one, then another, without replacement.

The big picture

A tree diagram shows a two-stage experiment as branches. Each end is one outcome. Multiply along a path to get that outcome's probability, and add the ends when more than one path gives what you want. Without replacement — including taking two at once — the second-stage fractions change because the bag has changed.

Key points

1Ends of a tree are mutually exclusive: only one path can happen, which is why you can add them.
2Events are independent when one doesn't change the chance of the other; on a tree, the second-stage branches are then the same whatever happened first.
3For two stages with a and b possible results, there are a × b outcomes to list.
4A two-way grid lists two-stage outcomes too, but a tree also handles more stages and unequal or changing probabilities.
5Check your tree: all the ends together add up to 1.

Worked example

Problem

The probability that a bus is late on any day is 0.2. Whether it is late one day has no effect on the next day. Draw a tree for Monday and Tuesday and find the probability that the bus is late on exactly one of the two days.

⚠ Watch out

Keeping the same fractions for the second pick when the first item isn't put back. If a red marble has gone, there is one fewer red AND one fewer marble altogether — change the top and the bottom.

🧠

Memory hook

Along a branch? Times it. Across the ends? Add it. And before the second pick, look in the bag again.

✓

Check yourself

3 green and 5 yellow counters are in a bag. Two are taken without replacement. What is P(both the same colour)? (Answer: 6/56 + 20/56 = 26/56 = 13/28.)

Flashcards

(11)
What does each end of a tree diagram show?
One complete outcome. Together, the ends list every possible outcome exactly once.
Red, then red: what do you do with the two branch fractions on that route?
Times them together. Both picks have to happen, so the fractions are multiplied.
When do you add probabilities on a tree?
When more than one end gives what you want. Find each path's probability, then add them — the ends can't both happen.
What must the branches from one point add up to?
1. Between them they cover everything that can happen at that stage.
What does 'mutually exclusive' mean?
The events can't both happen at the same time — like two different ends of the same tree.
What does 'independent' mean?
One event happening doesn't change the probability of the other.
With replacement: what happens to the second-stage branches?
Nothing changes. The item goes back, so the second pick has the same probabilities as the first — the picks are independent.
Without replacement: what happens to the second-stage fractions?
The count of the colour you took drops by 1, and so does the total. The second pick depends on the first.
How do you handle two items taken 'at once'?
Treat it as one item, then another, without replacement.
A first stage has 2 possible results and a second stage has 3. How many outcomes are there?
2 × 3 = 6 — the number of ends on the tree.
When is a two-way grid enough, and when do you need a tree?
A grid works for two stages with equally likely outcomes. Use a tree for more stages, or when the branches have different or changing probabilities.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

More AQA GCSE Maths topics

How this lesson was checked. This AQA GCSE Maths (specification 8300)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 29 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.