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Probability Basics - Edexcel GCSE Higher Maths

Typically 3-6 marks per paper, usually split across two short questions · Spec 1MA1

Probability basics is the set of ideas everything else in the strand is built on: a probability is a number from 0 to 1, the probabilities of an exhaustive set of outcomes add up to 1, and multiplying a probability by the number of trials gives an estimate of how often something will happen. On Edexcel's 1MA1 specification this covers P1 to P4, which deal with recording outcomes in tables, using randomness and fairness to calculate expected outcomes, relating expected frequencies to theoretical probability on the 0 to 1 scale, and the sum-to-one property for mutually exclusive events. P7 belongs here too: constructing possibility spaces for single and combined experiments, which in practice means the six-by-six grid for two dice. All of it is common content, so a Foundation candidate meets the same material.

Because it is crossover content, Edexcel places it in the first third of a Higher paper, often as the opening question or the one straight after it. The shapes recur: a table of probabilities with one cell blank and a follow-up asking for a number of trials, a two-way table of counts with a probability to write down, or two dice with a total to find. Marks run from 1 for a single reading up to 3 when the missing probability has to be split between two outcomes first. Any of the three papers can carry it, and the calculator makes almost no difference: on Paper 1 you leave the answer as a fraction such as 536\frac{5}{36}, on Papers 2 and 3 you can key the decimal, and the arithmetic is short either way.

These marks are quick, which is exactly why they get dropped. The two biggest losses are dividing by the probability when you should multiply, and answering with a probability when the question asked for a number of times. Write the units of your answer into the working: 'about 76 times' rather than a bare 0.19. Then practise the sample space for two dice until you know that there are 36 outcomes and not 11, and check every probability you write is a number between 0 and 1 before you move on.

Worked example

A biased four-sided spinner can land on A, B, C or D. The probability that the spinner lands on A is 0.28. The probability that the spinner lands on B is 0.34. The probability that the spinner lands on C is equal to the probability that the spinner lands on D. Rosa is going to spin the spinner 400 times. Work out an estimate for the number of times the spinner will land on C.
[4 marks]
  1. The four outcomes are all that can happen, so their probabilities add to 1. That leaves 1−(0.28+0.34)=0.381 - (0.28 + 0.34) = 0.38 to share between C and D.
    M1: subtracting the two given probabilities from 1
  2. C and D are equally likely, so each takes half of what is left: P(C)=0.38÷2=0.19P(C) = 0.38 \div 2 = 0.19.
    M1: halving the remainder, not assigning 0.38 to C
  3. An estimate of how often an outcome happens is its probability multiplied by the number of trials: 0.19×4000.19 \times 400.
    M1: probability times number of trials, the right way round
  4. 0.19×400=760.19 \times 400 = 76, so the spinner is expected to land on C about 76 times.
    A1: 76; a final answer of 0.19 scores the method marks only

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Nothing is printed as a picture, so tables of probabilities and two-way tables of counts arrive as a few lines of text, and every spinner and dice says how many sides it has and whether it is fair. Type probabilities as fractions if you prefer (7/36 is fine); where the exact value does not terminate the stem asks for a decimal to 3 decimal places instead.

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 Exam pace [1 mark]
    A fair six-sided dice numbered 1 to 6 is rolled once. Which pair of events are mutually exclusive?
    Answer options for question 1
  2. Question 2 Exam pace [2 marks]
    Amy, Ben, Cara and Dev belong to a club. Two of the four are chosen at random to be joint captains. Work out the probability that Amy is one of the two chosen.
  3. Question 3 Exam pace [2 marks]
    A cafe recorded what 200 customers bought one morning. 64 bought tea, 88 bought coffee and 32 bought hot chocolate. The rest bought no drink. Each customer bought at most one drink. One of the 200 customers is chosen at random. Work out the probability that this customer bought tea or hot chocolate.
  4. Question 4 Exam pace [3 marks]
    Two fair six-sided dice, each numbered 1 to 6, are rolled. Work out the probability that the difference between the two scores is 1. Give your answer as a fraction, or as a decimal to 3 decimal places.
  5. Question 5 Stretch [4 marks]
    Two fair six-sided dice, each numbered 1 to 6, are rolled and the two scores are added. Work out the probability that the total is a prime number. Give your answer as a fraction, or as a decimal to 3 decimal places.
  6. Question 6 Stretch [3 marks]
    Two fair six-sided dice, each numbered 1 to 6, are rolled together 180 times. Each time, the two scores are added. Work out an estimate for the number of times the total will be 8.

Common mistakes examiners see

  • Dividing by the probability instead of multiplying by it. With P(six)=0.2P(\text{six}) = 0.2 and 240 rolls, the answer comes out as 240÷0.2=1200240 \div 0.2 = 1200, which is more sixes than there are rolls.

    Expected number equals probability times number of trials. Test the size of your answer against the number of trials: an outcome with probability 0.2 must happen fewer than 240 times, so anything above 240 is wrong before you check the arithmetic.

  • Finding the missing probability and stopping there. The table is completed correctly with 0.27 in the blank cell, but the question asked how many of the 200 items would be pink, so 0.27 earns the method mark and nothing else.

    Underline what the last line of the question actually asks for. If the words are 'how many', 'estimate the number of' or 'work out how many times', your answer is a whole number of things, not a decimal between 0 and 1.

  • Using 11 or 12 as the denominator for two dice because there are 11 possible totals, so P(total=7)P(\text{total} = 7) is written as 111\frac{1}{11} rather than 636\frac{6}{36}. The same slip counts (2, 5) and (5, 2) as one outcome.

    Two dice give 6×6=366 \times 6 = 36 equally likely outcomes, and the two dice are separate objects, so the pairs (2, 5) and (5, 2) are both in the grid. Build the six-by-six table of totals once in practice and the counts for each total stop needing to be worked out from scratch.

  • Adding the probabilities of two events that can both happen. If P(A)=0.6P(A) = 0.6 and P(B)=0.5P(B) = 0.5, writing P(A or B)=1.1P(A \text{ or } B) = 1.1 gives an impossible probability.

    The addition rule P(A)+P(B)P(A) + P(B) only works when the events are mutually exclusive, meaning they cannot both occur. Otherwise use P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B), which is printed on the Exam Aid issued with every 1MA1 paper. Any probability above 1 is a signal to go back, not to round down.

  • Taking the denominator from a row or a column of a two-way table when the person is chosen from everybody, so 25 out of a column total of 55 is written instead of 25 out of 80. Presenting the answer as '25 out of 80' or as the ratio 25 : 80 also loses the mark.

    Read the sentence that starts 'One of the ... is chosen at random' and take the denominator from it. Write the answer as a fraction, a decimal or a percentage; a ratio is not a probability and Edexcel does not accept it.

Frequently asked questions

How do you work out the expected number of times something will happen?
Multiply the probability of the outcome by the number of trials. A spinner with a 0.19 chance of landing on green, spun 400 times, is expected to land on green about 0.19 times 400, which is 76 times. The word expected does not mean guaranteed, so Edexcel marks the answer as an estimate and phrases the question as 'work out an estimate for the number of times'.
How many outcomes are there when you roll two dice?
36, because each of the 6 scores on the first dice can pair with each of the 6 scores on the second. The two dice count as separate objects, so a 2 then a 5 is a different outcome from a 5 then a 2. The 11 possible totals from 2 to 12 are not equally likely, which is why 36 rather than 11 is the denominator.
Is this topic on the Foundation paper as well as Higher?
Yes. Spec points P1 to P4 and P7 are common to both tiers, so the probability scale, the sum-to-one property, expected frequency and possibility spaces appear on Foundation papers too. On a Higher paper they turn up early and are worth 1-3 marks each, and the Higher-only work sits further along the strand in conditional probability.
Do I get any probability formulae in the exam?
The Exam Aid that Pearson issues with each 1MA1 paper prints P(A or B) = P(A) + P(B) - P(A and B), and the Higher version also prints P(A and B) = P(A given B) P(B). What is not printed is the sum-to-one property or the expected frequency calculation, so those two you carry in your head.

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