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Tree Diagrams - Edexcel GCSE Higher Maths

Typically 4-6 marks in one question, on any of the three papers · Spec 1MA1

A probability tree diagram sets out two or more successive events as branches, with the probability of each outcome written along the branch it belongs to. On Edexcel's 1MA1 specification it sits under P6 (enumerating sets and combinations systematically using tables, grids, Venn diagrams and tree diagrams) and P8 (the probability of independent and dependent combined events). Both tiers meet tree diagrams. What belongs to Higher is selection without replacement, three-stage trees, and the questions that finish with an equation to solve for the number of counters in the bag.

Edexcel usually builds the question in parts. Part (a) hands you a half-drawn tree and asks for the missing probabilities, worth 1-2 marks. Part (b) wants a single path, so one product, worth 2 marks. Part (c) is where candidates separate: 'at least one', 'exactly one' or 'both the same colour', where two or three paths have to be found and then added, usually 3 marks. It can appear on any of the three papers. On Paper 1 you are handling fractions without a calculator, which is why mark schemes are full of unsimplified values like 3056\frac{30}{56}; on Papers 2 and 3 the probabilities are more often decimals and the danger shifts to keying the product in correctly.

Multiply along a path, add between paths. Most of the lost marks come from four places: keeping the first-stage denominator on the second stage when the counter is not replaced, adding two probabilities that should have been multiplied, reading 'at least one' as 'exactly one' and dropping the both-happen case, and finding red-then-blue while forgetting blue-then-red. Write down which paths you need before calculating anything, and reach for 1−P(none)1 - P(\text{none}) every time the question says at least.

Worked example

A bag contains 5 blue counters and 3 yellow counters. Joseph takes a counter at random and does not replace it. He then takes a second counter at random. Work out the probability that he takes one counter of each colour.
[4 marks]
  1. First stage: there are 8 counters, so P(blue)=58P(\text{blue}) = \frac{5}{8} and P(yellow)=38P(\text{yellow}) = \frac{3}{8}. The counter is not replaced, so only 7 counters remain for the second pick.
    M1: correct first-stage probabilities, with a denominator of 7 on the second stage
  2. Second stage: if the first counter was blue, 4 blue and 3 yellow are left, so P(yellow second)=37P(\text{yellow second}) = \frac{3}{7}. If the first was yellow, 5 blue and 2 yellow are left, so P(blue second)=57P(\text{blue second}) = \frac{5}{7}.
    M1: second-stage probabilities that change with the branch taken
  3. Multiply along each path: 58×37=1556\frac{5}{8} \times \frac{3}{7} = \frac{15}{56} for blue then yellow, and 38×57=1556\frac{3}{8} \times \frac{5}{7} = \frac{15}{56} for yellow then blue.
    M1: a correct product for at least one of the two mixed-colour paths
  4. Blue then yellow OR yellow then blue, so add the two paths: 1556+1556=3056=1528\frac{15}{56} + \frac{15}{56} = \frac{30}{56} = \frac{15}{28}.
    A1: both paths added, 1528\frac{15}{28} (0.536 to 3 decimal places)

Practice questions

These are original questions written in Edexcel 1MA1 Higher style. There are no printed trees here, so every bag, box and drawer is described in words: read for how many of each item there are and whether the first one goes back before the second is taken. Type answers as fractions (7/15 is fine) or, where the stem asks, as a decimal to 3 decimal places.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [1 mark]
    The probability that Ella's train is late on any morning is 0.18. Write down the probability that her train is not late on a given morning.
  2. Question 2 Exam pace [3 marks]
    A packet contains 6 lemon sweets and 4 mint sweets. Jo takes a sweet at random and eats it, then takes a second sweet at random and eats it. Work out the probability that she eats one sweet of each flavour. Give your answer as a fraction or as a decimal to 3 decimal places.
  3. Question 3 Exam pace [4 marks]
    A darts team plays a match on Saturday and a match on Sunday. The probability that the team wins on Saturday is 0.45. If the team wins on Saturday, the probability that it wins on Sunday is 0.6. If the team does not win on Saturday, the probability that it wins on Sunday is 0.3. Work out the probability that the team wins exactly one of the two matches.
  4. Question 4 Exam pace [3 marks]
    Dan passes two sets of traffic lights on his way to work. The probability that the first set is green when he reaches it is 0.4. The probability that the second set is green when he reaches it is 0.55. The two sets of lights work independently. Work out the probability that at least one set is green when he reaches it.
  5. Question 5 Exam pace [3 marks]
    A club has 20 members: 12 girls and 8 boys. Two different members are chosen at random to represent the club, one after the other, so nobody can be chosen twice. Work out the probability that both are girls. Give your answer as a fraction or as a decimal to 3 decimal places.
  6. Question 6 Stretch [4 marks]
    A bag contains 3 red counters, 4 blue counters and 5 green counters. Two counters are taken at random, one after the other, without replacement. Work out the probability that the two counters are the same colour. Give your answer as a fraction or as a decimal to 3 decimal places.

Common mistakes examiners see

  • Leaving the second-stage probabilities unchanged when the first item is not replaced, so 4 red counters out of 10 give 410×310\frac{4}{10} \times \frac{3}{10} (0.12) rather than 410×39\frac{4}{10} \times \frac{3}{9} (0.133 to 3 decimal places).

    Before writing anything on the second set of branches, say out loud what is left in the bag: one fewer of the colour you took, one fewer in total. The second-stage denominator is always one less than the first.

  • Adding along a path instead of multiplying, so P(both events happen) comes out as 0.4+0.3=0.70.4 + 0.3 = 0.7 instead of 0.4×0.3=0.120.4 \times 0.3 = 0.12.

    Along the branches means AND, so multiply. Between finished paths means OR, so add. Sanity check the size: multiplying two probabilities must give an answer smaller than either of them.

  • Treating 'at least one' as 'exactly one', adding only the two mixed paths and losing the case where the event happens twice.

    Use 1−P(neither)1 - P(\text{neither}). That is one product rather than two or three, and it cannot leave a case out. Underline the words 'at least' in the stem before you start, because this is the single most common omission on Edexcel tree diagram parts.

  • Working out red then blue and stopping, when 'one of each colour' also allows blue then red, which halves the answer.

    Write the list of paths that match the description before you calculate. If the question names two outcomes without fixing the order, there are two paths and both belong in the sum.

  • Cancelling each product as you go, then trying to add 514\frac{5}{14} and 328\frac{3}{28} under time pressure, or slipping to 0.2×0.2=0.40.2 \times 0.2 = 0.4 on the calculator paper.

    Keep the products over the same denominator (2056\frac{20}{56} and 656\frac{6}{56}) and simplify once, at the end. On Papers 2 and 3 type the whole product into the calculator in one line instead of multiplying two decimals in your head.

Frequently asked questions

Are tree diagrams on the Foundation paper as well as Higher?
Yes. Spec points P6 and P8 are on both tiers, so Foundation candidates use tree diagrams too, usually with two independent stages and the item replaced. Higher adds selection without replacement, three-stage trees, and questions that end with an equation to solve for the number of items in the bag.
Do you add or multiply on a probability tree diagram?
Multiply along a path, because that is one outcome AND then another. Add between completed paths, because those are separate ways of reaching the result. Two checks: the probabilities on any one set of branches add to 1, and the probabilities at the ends of all the paths also add to 1.
How many marks are tree diagram questions worth on Edexcel Higher?
Filling in missing branch probabilities is normally 1-2 marks, a single product is 2 marks, and an 'at least one' or 'exactly one' part is usually 3 marks. A whole question therefore runs to about 4-6 marks, and it can be set on Paper 1, 2 or 3.
Do I have to draw the tree diagram to get the marks?
No. The marks are for correct probabilities and correct products, and Edexcel frequently prints the tree for you. Drawing one anyway is the quickest way to see every path, and it gives the examiner working to award a method mark to when the final answer is wrong.

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