Probability — AQA GCSE Mathematics
Test yourself on Probability with AQA GCSE practice questions.
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Probability explained
A frequency tree splits a group of people or trials by one feature and then by a second.
Read the full explanation
Every branch carries a whole number, not a probability, and each fan of branches adds back to the total above it. Fill one in by starting from the number you are given and subtracting to find its partner. For example, of 80 patients, 30 are children; 12 children and 20 adults test positive. The child branch splits into 12 and 18, and the adult branch, holding 50 patients, splits into 20 and 30. A two-way table holds the same information in rows and columns, with totals down the side and along the bottom. Once the diagram is complete, a probability is a branch frequency over the relevant total.
apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
Expected frequency is a prediction made before an experiment happens: multiply the probability of an outcome by the number of trials. If a fair spinner has five equal sectors and two of them are red, P(red) = 2/5, so in 200 spins you expect 2/5 of 200, which is 80 reds. Fair means every outcome is equally likely, so with n equally likely outcomes each has probability 1/n. Biased means they are not equal. An expected value is what you would bet on, not what must occur, so getting 78 reds does not prove the spinner is unfair. Questions often give a probability, or a relative frequency from a trial run, and ask you to scale it up to a larger number of future trials.
relate relative expected frequencies to theoretical probability, using appropriate language and the 0 to 1 probability scale
Every probability sits between 0, meaning impossible, and 1, meaning certain, and can be marked on a scale line as a fraction, decimal or percentage. Words such as unlikely, even chance and very likely say roughly where a value sits, so use one of them when a question asks you to describe a chance, and give a number whenever it asks you to work one out. A theoretical probability comes from counting equally likely outcomes: on a fair six-sided dice, P(more than 4) = 2/6 = 1/3. An experimental probability, also called relative frequency, comes from results you actually got, as successes divided by trials. A common question gives a table of trial results, asks for the relative frequency of one outcome, then asks you to compare it with the theoretical value and say what that suggests.
apply the property that the probabilities of an exhaustive set of outcomes sum to 1 apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1
A set of outcomes is exhaustive when it covers everything that can happen, and two events are mutually exclusive when they cannot both happen at once. When both conditions hold, the probabilities total 1, and that gives you two moves. First, find a missing value by subtracting the others from 1: if P(red) = 0.3 and P(blue) = 0.45, then P(green) = 1 − 0.3 − 0.45 = 0.25. Second, use the complement, P(not A) = 1 − P(A), whenever the question asks for the chance that something does not happen. Tables often hide an algebraic step, with two cells both labelled x, so you form an equation such as 0.3 + 2x = 1 and solve it. Every value you write should end up between 0 and 1.
understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
Repeat a fair experiment more times and the relative frequency of an outcome tends to settle closer to its theoretical probability. Twenty spins of a fair coin might give 13 heads, a relative frequency of 0.65, while two thousand spins would be expected to sit far nearer 0.5. That is why a question asking which estimate is best points you to the row with the most trials, not to the value that happens to look neatest. Two conditions matter: the sample must be unbiased, so nothing about how it was taken favours one outcome, and it must be large. Small samples vary a lot, so a short run of odd results is not on its own evidence that a coin or a dice is unfair.
enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams
A list of outcomes only earns marks if it is complete, so work in a fixed order: hold the first item still, run through every partner, then move on. For two dice, a grid of six rows and six columns shows all 36 pairs at once. A Venn diagram sorts members by the properties they have. Fill the overlap first, subtract to find each single region, and put anything with neither property outside the circles. If a class of 30 has 18 studying French, 14 studying German and 5 studying both, the French-only region holds 13, the German-only region holds 9, and 3 pupils sit outside. The notation A ∩ B means the members in both sets, and A ∪ B means the members in one set or the other or both.
including using tree diagrams
A tree diagram sets out an experiment stage by stage, and one path read from left to right is one complete outcome. Draw the first stage, then copy that fan of branches onto the end of every branch already there, so nothing is missed. Copy it unchanged only while the same outcomes are still available: if something is taken away and not replaced, the later fans drop the outcomes that can no longer happen. Two stages of two outcomes give four paths, and two stages of three outcomes give nine. Label the end of each path with the outcome it names, for example HH, HT, TH and TT for two coins. Trees suit stages that are not identical, such as choosing a shirt and then a tie, and a third stage means fanning out once more. When probabilities are added to the branches, the numbers on any one fan add to 1 and a path is followed by multiplying.
construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
A possibility space is the full set of what can happen, best drawn as a grid when two things happen together. Put the outcomes of one experiment along the top, the other down the side, and fill each cell with the combined result. For the total of two fair dice the grid has 36 cells, and the totals inside them run from 2 up to 12. Because the cells are equally likely, a probability is the number of cells that fit over the number of cells altogether: P(total of 5) = 4/36 = 1/9. Equally likely is the condition that makes this work, so it applies to fair dice, coins and spinners. Check whether the cells should hold sums, differences, products or ordered pairs, since the entries change completely.
calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
Combined events can be independent or dependent. Two events are independent if one does not affect the other, e.g., two coin flips. For these, the multiplication rule is P(A and B) = P(A) × P(B). Events are dependent if the first outcome changes the probability of the second, such as drawing counters from a bag 'without replacement'. Representations include tree diagrams, two-way tables, and Venn diagrams. A tree diagram is useful for sequential events. For a bag with 4 red and 6 blue counters, P(Red then Red) = (4/10) × (3/9) = 12/90, as the second pick is from 9 counters. A two-way table is ideal for showing all outcomes from two independent events, like rolling two dice, which gives 36 cells. The key assumption is often randomness (e.g., a fair coin or random selection). For 'at least one' questions, calculate 1 − P(none).
calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams (Higher tier only)
This is Higher tier only. A conditional probability asks how likely one event is once you already know another has happened, and the effect is to shrink the group you are choosing from. Written P(A given B), it is the number that are in both, divided by the number in B altogether, which includes those who are in A as well. Expected frequencies make it concrete, so fill a two-way table or a Venn diagram with counts and read the answer straight off. If 60 people are surveyed, 25 own a dog and 10 of those dog owners also own a cat, then P(cat given dog) = 10/25 = 2/5, because only the dog owners are still in play. On a tree diagram the same idea is the second set of branches, whose values depend on the branch before them.
Your focus
- Write down what each branch of a frequency tree holds, explaining why the entries are whole numbers rather than probabilities.
- Complete a frequency tree or two-way table by subtracting a known branch frequency from the total above it to find its partner.
- Work out a probability from a completed tree or table as a branch frequency over the relevant total.
Show all 35 objectives
- Describe a feature of the data read off a completed table, quoting the frequencies rather than asserting the claim.
- Explain what fair means, stating that each of n equally likely outcomes has probability 1/n while biased means they are not equal.
- Calculate an expected frequency by multiplying the probability by the number of trials, so 2/5 of 200 spins predicts 80 reds.
- Explain why getting 78 reds in 200 spins does not prove bias, treating an expected value as a prediction not a guarantee.
- Describe a chance in words such as unlikely or even chance, and mark a given probability in the right place on a 0 to 1 scale.
- Work out a relative frequency as successes divided by trials, and a theoretical probability by counting equally likely outcomes.
- Compare an experimental probability with the theoretical value, quoting both figures and saying which is larger and what it suggests.
- Explain what exhaustive and mutually exclusive mean, and state that the probabilities of such a set of outcomes total 1.
- Calculate a missing probability by subtracting the given values from 1, and find P(not A) as 1 minus P(A).
- Form and solve an equation such as 0.3 + 2x = 1 when a table hides an unknown, checking each value lies between 0 and 1.
- Explain why the relative frequency of an outcome settles closer to its theoretical probability as the number of trials grows.
- Work out the best estimate of a probability from combined totals across several sets of trials rather than from one set.
- Justify which of several estimates is most reliable by naming the number of trials, not how neat the values look.
- Explain why a short run of odd results is not evidence of bias, referring to how much small samples vary.
- Write down what A and B in both sets means and what A or B means, using the intersection and union notation.
- Complete a Venn diagram by placing the overlap first, then subtracting it from each set total and putting the rest outside.
- Work out a probability from a completed grid or Venn diagram, using the correct total as the denominator.
- Explain why a list of outcomes must follow a fixed order, holding the first item still while running through every partner.
- Describe how a tree diagram is built stage by stage, with each fan of branches copied onto the end of every branch already drawn.
- Draw a tree for two or three stages and label each path end with its outcome, such as HH, HT, TH and TT.
- Explain how a later fan changes when an item is not replaced, dropping the outcomes that can no longer happen.
- Work out an answer from a completed tree by counting the paths that meet the condition, checking each fan totals 1.
- Complete a possibility space grid for two combined experiments, checking whether the cells hold sums, differences, products or pairs.
- Calculate a theoretical probability as favourable cells over total cells, so P(total of 5) on two dice is 4/36 or 1/9.
- Explain why equally likely outcomes are the condition that lets you count cells, so the method fits fair dice, coins and spinners.
- Explain the difference between independent and dependent events, naming without replacement as the signal that the second stage changes.
- Complete the second stage of a dependent tree with a denominator one less than the first, such as 3/9 red after a red is taken.
- Calculate a combined probability by multiplying along each path and adding the paths that fit, as in 4/10 times 3/9 giving 2/15.
- Justify using 1 minus P(none) for an at least one question, and state the assumption the calculation relies on.
- Explain that a conditional probability shrinks the group being chosen from, so only the members of the given event remain in play.
- Calculate a conditional probability from expected frequencies in a two-way table or Venn diagram, as in 10 out of 25 dog owners.
- Interpret a conditional probability in the context of the question, saying what it tells you about the restricted group.
Probability exam tips
Quick Revision Summary (Key Takeaway)
Probability is the branch of mathematics that measures how likely an event is to occur, expressed as a number between 0 and 1 on a probability scale. At AQA GCSE, students must calculate theoretical and experimental probabilities, use the addition and multiplication rules for combined events, and work with tree diagrams, Venn diagrams and two-way tables.
Topic Overview
Probability is the study of chance and uncertainty, forming a core part of the AQA GCSE Mathematics specification. It covers theoretical probability, experimental probability, the probability scale from 0 to 1, and the rules for combining events such as the addition rule and multiplication rule. Students also learn to represent probability problems using tree diagrams, Venn diagrams, and two-way tables.
This topic is essential for interpreting real-world data, making informed decisions, and solving problems in science, economics, and everyday life. It frequently appears in both Foundation and Higher tier exams, often combined with ratio, fractions, and algebra. Mastery of probability also builds a foundation for A-level Mathematics and further statistical study.
Key Concepts
- →The probability scale: all probabilities lie between 0 (impossible) and 1 (certain), and the sum of probabilities of all mutually exclusive outcomes is 1.
- →Theoretical probability = number of favourable outcomes / total number of equally likely outcomes.
- →Experimental probability = relative frequency = number of successful trials / total number of trials; it may differ from theoretical probability but tends to get closer with more trials.
- →Addition rule: P(A or B) = P(A) + P(B) - P(A and B) for non-mutually exclusive events; for mutually exclusive events, P(A or B) = P(A) + P(B).
- →Multiplication rule: P(A and B) = P(A) x P(B) for independent events; for dependent events, P(A and B) = P(A) x P(B|A).
Marking Points
- each correct pair of frequencies filled into the tree, so one wrong entry does not lose everything
- a subtraction that recovers a missing frequency from the total above it, even if the final answer is wrong
- a probability written as a frequency over the correct total, accepted as a fraction, a decimal or a percentage
- a described feature of the data read off the completed table rather than asserted
- the probability of the single outcome, in any correct form as a fraction, decimal or percentage
- multiplying that probability by the number of trials, even if the probability itself is wrong
- an expected frequency given as a whole number of items where the context needs one
- an explanation of fairness that refers to the given figures rather than to opinion
- a relative frequency written as successes over the total number of trials, before any simplifying
- the theoretical probability of the same outcome, even if the comparison that follows is wrong
- a comparison that quotes both values and says which is larger, not only that they differ
- a probability marked in the right place on the scale, within the tolerance allowed
- the sum of the probabilities that are given, or for setting up the subtraction from 1
- an equation whose left-hand side totals every probability in the table, when an unknown appears
- solving that equation correctly, even if the equation came from a wrong total
- a final probability in an acceptable form, as a fraction, a decimal or a percentage
- choosing the estimate based on the greatest number of trials as the most reliable
- a reason that names the sample size, not only how close two values happen to be
- a relative frequency worked out from combined totals across several sets of trials, rather than from one set alone
- a conclusion about bias that compares the estimate with the theoretical probability
- a systematic list or grid showing every outcome once, with nothing repeated and nothing missing
- the overlap placed correctly in the centre of the Venn diagram before the other regions are filled
- each remaining region found by subtracting the overlap from a given set total
- a probability read off the completed diagram with the correct total as the denominator
- the correct number of branches at each stage, with the stages clearly separated
- outcomes labelled at the ends of the paths in the order the stages happen
- a complete set of paths with none repeated, even if a later calculation is wrong
- using the completed tree to answer the question, such as counting the paths that meet a condition
- a grid or list showing all the possible outcomes of the combined experiment
- identifying the outcomes that meet the condition, for example by ringing those cells
- a probability written as successful outcomes over total outcomes, even if the count is wrong
- the final probability, accepted as a fraction, a decimal or a percentage unless the question asks for one particular form
- For a single fair die, list the sample space {1, 2, 3, 4, 5, 6}; two outcomes exceed 4, so the probability is 2/6 = 1/3.
- Correctly setting up a two-way table or Venn diagram with given information.
- Calculating second-stage probabilities on a dependent tree diagram (e.g., denominator reduced by one).
- Multiplying probabilities along the branches of a tree diagram to find the probability of a combined outcome.
- Adding the probabilities of all required final outcomes (e.g., P(Red then Blue) + P(Blue then Red)).
- Stating a relevant assumption, such as events being random or independent, when asked.
- a two-way table or Venn diagram completed with expected frequencies rather than probabilities
- using the restricted group as the denominator, even if the numerator is wrong
- the conditional probability written as a fraction of that restricted group
- interpreting the value in the context of the question when asked to comment on it
Examiner Tips
- 💡Write the grand total at the start of the tree before anything else, because every later subtraction depends on it.
- 💡Check that each set of branches adds to its parent total before using the frequency tree; inconsistent totals can lead to incorrect probabilities.
- 💡Leave the completed frequencies on the printed diagram, as the filled tree itself usually carries marks.
- 💡Work out the probability of one trial first, then multiply by the number of trials, in that order.
- 💡Give the exact value from the multiplication as your answer, and round to a whole number only when the question asks for a number of people or objects.
- 💡Check whether the question says fair or biased before assuming the outcomes share the probability equally.
- 💡Take the denominator from the total row of the table, not from the biggest number in it.
- 💡Give a number when the question asks you to work out, calculate or estimate a probability, and keep words such as likely for when it asks you to describe the chance.
- 💡Keep the unsimplified fraction next to the simplified one so the method stays visible.
- 💡Add the probabilities you are given first, because the gap up to 1 is usually the answer or the next step.
- 💡Convert everything to the same form, all decimals or all fractions with a common denominator, before adding.
- 💡If a value comes out negative or above 1, go back through the working, since a probability can be neither.
- 💡When a table shows results after 10, 50, 100 and 500 trials, quote the estimate from the largest total.
- 💡Put the words more trials into any explanation about reliability, as that is the idea being credited.
- 💡If the trials in a table are running totals, use those totals rather than treating each block as a fresh experiment.
- 💡Start a Venn diagram with the number in both sets, because every other region is worked out from it.
- 💡Count the entries in your list against a quick calculation, such as six times six for two dice, to check none are missing.
- 💡Write the number with neither property outside the circles, since it still belongs to the total.
- 💡Draw the branches with a ruler and keep each fan the same shape, because outcomes go missing in a cramped tree.
- 💡Count the paths as a check: it is the number of options at each stage multiplied together.
- 💡Keep the outcomes in the same order at every stage so the finished tree is quick to read.
- 💡Draw the grid even when the question does not ask for one, because counting from a grid beats counting in your head.
- 💡Write the total number of cells down before counting the successful ones, so the denominator is settled.
- 💡Test a corner entry against its row and column headings before you rely on the rest of the grid.
- 💡For sequential events, especially 'without replacement', a tree diagram is usually best.
- 💡For two simultaneous independent events (e.g., rolling two dice, spinning two spinners), a two-way table is often clearer and quicker.
- 💡To find P(at least one...), it is often easier to calculate 1 − P(none...).
- 💡Always check that the probabilities on any set of branches from a single point on a tree diagram add up to 1.
- 💡Underline the words given that, and ring the group they restrict you to, before writing anything down.
- 💡If only probabilities are supplied, turn them into frequencies out of a convenient total such as 100, because whole numbers make conditional work far easier.
- 💡Check the denominator is smaller than the whole sample, since using the grand total almost always means the wrong question has been answered.
- 💡Always show your working, especially when using tree diagrams or formulas. Method marks are awarded even if the final answer is wrong.
- 💡Read the question carefully to determine whether events are independent or dependent, and whether they are mutually exclusive. This decides which rule to use.
- 💡For probability questions involving fractions, simplify your final answer unless the question asks for a decimal or percentage. Check that your answer is between 0 and 1.
Common Mistakes
- putting probabilities on the branches of a frequency tree when the question asks for frequencies, so the numbers stop adding to the total
- subtracting from the grand total instead of from the branch total directly above the gap
- reading the denominator off the wrong branch, so a probability is taken out of the whole group when only one branch was meant
- dividing the number of trials by the probability, or adding the two, instead of multiplying
- treating outcomes as equally likely when the question supplies a biased spinner or a table of unequal frequencies
- saying an experiment has gone wrong because the real count missed the expected value by a small amount
- giving an answer above 1, or as a ratio such as three to seven, when a probability must lie on the scale from 0 to 1
- dividing by the number of successes rather than by the number of trials when finding relative frequency
- reading the largest frequency in a table as the denominator instead of the total
- adding probabilities of events that can happen together, such as drawing a red card and drawing a king, as though they were mutually exclusive
- making a table of decimals total 100, or mixing percentages with decimals in the same sum
- subtracting only one of the given probabilities from 1 when several are given
- picking the estimate nearest the theoretical value as the best one instead of the estimate from the largest sample
- averaging or adding relative frequencies from several rows instead of combining the raw frequencies and the raw totals
- declaring a coin biased because a short run of trials gave an uneven split
- writing a whole set total inside one region of a Venn diagram, so the regions add up to more than the number of members
- listing pairs at random and missing one, for example giving the outcomes of two coins as head-head, head-tail and tail-tail
- confusing union with intersection, so the overlap alone is shaded when the question asked for everything in either set
- drawing the second stage on the first branch only, so half the outcomes never appear
- labelling a path in the wrong order, so red then blue and blue then red are treated as the same outcome
- repeating the first stage labels on the second stage of a without-replacement experiment, which lists a path that cannot happen
- using the number of different totals as the denominator, so a two-dice question is worked out over 11 rather than over 36
- counting an ordered pair once when both orders are possible, such as treating a three with a five as a single outcome
- filling the grid with the outcomes of one experiment only, so the combined results are never formed
- Forgetting to reduce the denominator on the second branch of a 'without replacement' tree diagram.
- Adding probabilities along tree diagram branches instead of multiplying.
- Miscounting the total number of outcomes in a sample space diagram (e.g., 6+6=12 for two dice instead of 6×6=36).
- Assuming events are independent when they are dependent, and using P(A) × P(B) incorrectly.
- dividing by the overall total instead of by the total for the event you are told has happened
- reversing the condition, working out the chance of the dog given the cat when the question asked for the cat given the dog
- quoting the number in both categories on its own without dividing by the size of the given group
- Students often think that if an event has not happened for a while, it is 'due' to happen (gambler's fallacy). In reality, independent events remain unaffected by previous outcomes.
- Many students add probabilities when they should multiply, especially when dealing with 'and' situations. Remember: 'and' usually means multiply, 'or' usually means add (with adjustment for overlap).
- When using tree diagrams, students sometimes forget to adjust probabilities for the second event when sampling without replacement. Always check whether the total number of items has changed.
Revision Plan
- 1Day 1-2: Revise the basics: probability scale, theoretical and experimental probability, and the concept of mutually exclusive and independent events.
- 2Day 3-4: Practice using the addition and multiplication rules with simple examples. Focus on identifying when to add and when to multiply.
- 3Day 5-6: Learn to construct and interpret tree diagrams for both independent and dependent events. Practice at least five problems with and without replacement.
- 4Day 7-8: Work through Venn diagram and two-way table problems, including conditional probability and set notation.
- 5Day 9-10: Complete past paper questions on probability, timing yourself, and review any mistakes. Use the mark scheme to understand where marks are awarded.
Exam Question Types
- 📋Calculating simple theoretical probability from a given situation (e.g., dice, counters, cards). Advice: always write the probability as a fraction and simplify.
- 📋Using tree diagrams to solve problems involving two or more events, often with and without replacement. Advice: draw the diagram clearly, label branches, and show multiplication and addition steps.
- 📋Applying the addition rule for non-mutually exclusive events, often presented in Venn diagram or two-way table format. Advice: identify the overlap and subtract it to avoid double-counting.
- 📋Experimental probability and relative frequency questions, sometimes asking for expected outcomes. Advice: use the formula relative frequency = number of successes / total trials, and scale up for expected values.
Command Word Expectations (AQA)
Work out a numerical answer using the given information. You must show sufficient working to earn method marks, and give your answer in the required form (fraction, decimal, or percentage).
Prove a given result by showing each step of your working. You must include all relevant calculations and reasoning, and conclude with a statement that matches the given result.
Give reasons for your answer, referring to probability rules or concepts. You must use correct mathematical terminology and link your explanation to the context of the question.
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: A bag contains 5 red counters and 3 blue counters. A counter is taken at random and not replaced. A second counter is then taken. Calculate the probability that both counters are red.
- 1.Step 1: Identify given facts: 5 red, 3 blue, total 8 counters. First counter is not replaced.
- 2.Step 2: Apply the multiplication rule for dependent events: P(first red) = 5/8. After removing one red, there are 4 red and 3 blue left, total 7, so P(second red) = 4/7.
- 3.Step 3: Multiply the probabilities: P(both red) = 5/8 x 4/7 = 20/56 = 5/14.
- 4.Step 4: State final conclusion: The probability is 5/14 or approximately 0.357.
Question: In a class of 30 students, 18 study French, 12 study Spanish, and 6 study both. A student is chosen at random. Find the probability that the student studies French or Spanish.
- 1.Step 1: Identify given facts: P(F) = 18/30, P(S) = 12/30, P(F and S) = 6/30.
- 2.Step 2: Apply the addition rule for non-mutually exclusive events: P(F or S) = P(F) + P(S) - P(F and S).
- 3.Step 3: Substitute: P(F or S) = 18/30 + 12/30 - 6/30 = 24/30.
- 4.Step 4: Simplify: 24/30 = 4/5.
- 5.Step 5: State final conclusion: The probability is 4/5 or 0.8.