Probability

• Define the event, model every outcome, assign the probabilities, multiply along a route and add across routes, then check \(0\leq P\leq1\).
• Equally likely? Replaced or not (Extended)? Do they overlap? Does one change the other? Observed or theoretical? Five questions, and the model is decided.
• Every one of these nine is caught by the same two checks: does the answer lie between 0 and 1, and do my exhaustive outcomes total exactly 1? For a conditional answer, "exhaustive" means the options allowed by the condition — they must total 1 too.
• If you remember nothing else: define the event, model every outcome, multiply along, add across, and check the total is 1.
• If you remember nothing else on this page: without replacement, update the fraction; given that, change the denominator.

• C8.1 — place an event on the probability scale from 0 to 1, and say what a probability of 0 and a probability of 1 mean.
• C8.1 — calculate the probability of a single event by counting favourable outcomes out of equally likely outcomes, having first checked that the outcomes really are equally likely.
• C8.1 — give a probability as a fraction, a decimal or a percentage, with fractions simplified.
• C8.1 — use the rule that the probability an event does not happen is 1 minus the probability that it does, in both directions.
• C8.1 — extract a probability from a list, a two-way table, a graph or a two-set Venn diagram.
• C8.2 — calculate a relative frequency from experimental results and use it as an estimate of a probability.
• C8.2 — explain why a longer run of trials usually gives a more stable estimate, provided the conditions do not change.
• C8.2 — calculate an expected frequency as the number of trials multiplied by the probability, including estimating an expected value from a population, and say clearly why it is a prediction rather than a guarantee.
• C8.2 — use the terms fair, bias and random precisely, and judge fairness by comparing observed with expected frequencies while taking the number of trials into account.
• C8.3 — list the outcomes of a two-stage experiment systematically, and build a sample-space diagram or two-way outcome table without omission or duplication.
• C8.3 — complete a two-set Venn diagram from the intersection outwards, and read the "both", "only", "either" and "neither" counts from it.
• C8.3 — draw a tree diagram with outcomes at the ends of the branches and probabilities beside them, multiply along a route, and add alternative successful routes, for combined events with replacement.
• Across the topic — check that every probability you write lies between 0 and 1, that the probabilities out of any one node total 1, and that a set of exhaustive outcomes totals 1.
• E8.1 — read and write probability notation: \(P(A)\) for the probability of \(A\) and \(P(A')\) for the probability of not \(A\), including recovering \(P(B)\) from a given \(P(B')\).
• E8.1 — read a probability from a Venn diagram of three sets as well as two.
• E8.3 — use \(A\cap B\) and \(A\cup B\) with Venn diagrams, and apply \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), recognising the mutually exclusive case where \(P(A\cap B)=0\).
• E8.3 — handle selection without replacement, where both the numerator and the denominator of the second-stage probability change, and say why the answer differs from the with-replacement one.
• E8.4 — calculate a conditional probability from a two-way table, from a Venn diagram and from a tree diagram, by restricting the model to the part where the given condition holds — without using \(P(A\mid B)\) notation or any conditional-probability formula, neither of which is required.

A revision habit worth keeping past the exam. Whenever you meet a percentage in the news — a weather forecast, a medical risk, a poll — ask the two questions this chapter has been drilling: is that number theoretical or observed, and how many trials is it based on? Those two questions separate a reliable figure from a meaningless one, and they are the same two the examiner is testing.

Probability of Combined Events

A combined event involves more than one thing happening, either two features of the same object or two stages of an experiment, and its probability is calculated from a model of all the possible outcomes. Sample-space diagrams and two-way outcome tables list every ordered outcome of a two-part experiment, so that two fair dice give thirty-six equally likely ordered outcomes even though they produce only eleven possible totals, which are not equally likely. Venn diagrams organise overlapping sets and are completed from the intersection outwards, giving the intersection, the only regions, the neither region and the union, all of which must total the universal set; the Core route is limited to two sets while the Extended route is not. In the Extended route the addition rule states that the probability of A union B is the probability of A plus the probability of B minus the probability of A intersection B, so that the overlap is not counted twice, and when the events are mutually exclusive the intersection has probability zero and the rule reduces to simple addition; Core candidates reach the same answers by counting the regions of the diagram. Tree diagrams show successive stages with outcomes written at the ends of the branches and probabilities beside them, probabilities are multiplied along a single complete route and the products of alternative successful routes are added; Core combined events are always with replacement, so the second-stage probabilities are unchanged, while the Extended route also covers sampling without replacement, where both the numerator and the denominator are updated.

Relative and Expected Frequencies

Relative frequency is the observed frequency of an outcome divided by the number of trials, and it is used as an experimental estimate of a probability when the theoretical probability is unknown or when a device may not be fair. It is an estimate rather than a guarantee: repeating the experiment can give a different value, and a longer run of trials usually gives a more stable estimate provided the conditions do not change. Expected frequency is the number of trials multiplied by the probability of the outcome, and the same relationship is used to estimate an expected value from a population by multiplying the population size by the probability. An expected frequency is a prediction made by the model rather than a promise about what must occur. A device is fair when its intended outcomes have the appropriate equal theoretical probabilities and biased when they do not, while an outcome is random when an individual result cannot be predicted with certainty in advance even though its probabilities may be known. Fairness is judged by comparing observed with expected frequencies while taking the number of trials into account, because a short surprising run is not by itself proof of bias.

Introduction to Probability

Probability is a number from 0 to 1 that measures how likely an event is, where 0 means the event is impossible, 1 means it is certain, and values nearer 1 mean greater likelihood. In the Extended route the probability of event A is written P(A) and the probability that A does not occur is written P(A prime); probability notation is not required in the Core route, where the same ideas are stated in words. When every possible outcome is equally likely, P(A) is the number of favourable outcomes divided by the total number of possible outcomes, and the result may be given as a fraction, a decimal or a percentage, with fractional answers simplified. Because an event either occurs or does not occur, the two probabilities total 1, giving the complement rule that the probability an event does not occur is 1 minus the probability that it does, written P(A prime) equals 1 minus P(A) in the Extended route. The outcome information needed may come from a systematic list, a two-way table, a graph such as a bar chart, or a Venn diagram, which is limited to two sets in the Core route, and the counting method is valid only when the listed outcomes are genuinely equally likely.

Conditional Probability

A conditional probability is the probability of one event calculated on the understanding that another event is already known to have happened. It is found by restricting the model to the part in which the given condition holds and then treating that part as the whole: on a two-way table the row or column total named by the condition becomes the new denominator, on a Venn diagram the region named by the condition becomes the new universal set, and on a tree diagram the second-stage branch probabilities after a particular first-stage outcome are already conditional probabilities. When a condition covers several routes of a tree, the probabilities of the routes that satisfy the condition are added and the probability of the wanted routes is expressed as a share of that total. The result is generally different from the unconditional probability, and it is not symmetrical: the probability of A given B and the probability of B given A are usually different numbers. Knowledge of the notation P of A given B, and of formulas relating to conditional probability, is not required by the syllabus, so every calculation is carried out directly from the table, diagram or tree.

• 1. Counting outcomes that are not equally likely \(P(A)=\dfrac{\text{favourable}}{\text{total}}\) is a statement about equally likely outcomes only. Two dice have 36 equally likely ordered outcomes but only 11 possible totals, and those totals run from probability \(\frac{1}{36}\) up to \(\frac{6}{36}\). Answering "the total can be 2 to 12, so \(P(\text{total}=7)=\frac{1}{11}\)" is an expensive error, because it invalidates every answer built on top of it. Fix Before dividing, ask: is every outcome in my list as likely as every other? If not, go back down to the ordered outcomes underneath them.
• 2. Extended E8.3 Leaving the second denominator unchanged without replacement Extended only. Core combined events are always with replacement, so a Core candidate never meets this one — but every Extended tree question turns on it. If a counter is not put back, the bag is smaller. Both the numerator and the denominator of the second-stage probability may change, and which numerator changes depends on which branch you are on. Writing \(\frac{3}{5}\times\frac{2}{5}\) for a without-replacement question is not a small slip — it answers a different question. Fix Write the composition of the bag beside every node: "3R 2B" at the start, "2R 2B" after a red, "3R 1B" after a blue.
• 3. Adding overlapping events without allowing for the overlap Adding two probabilities is only safe when the two events cannot happen together. If they can, everything in the overlap has been counted twice, and the answer comes out too big — often greater than 1, which is the alarm. Fix Ask "can both happen at once?" before adding. If the answer is yes, draw the Venn diagram and count the regions instead: the overlap is then in the count exactly once, automatically. Extended E8.3 In notation, this is the addition rule: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), and \(P(A\cup B)=P(A)+P(B)\) holds only when \(P(A\cap B)=0\).
• 4. Confusing mutually exclusive with independent These are opposite kinds of statement. Mutually exclusive says the two events cannot both happen. Independent says one happening does not change the chance of the other. Two mutually exclusive events with non-zero probability are as far from independent as it is possible to be: if one happens, the other's probability drops straight to zero. Fix Exclusive is a question about the sample space — do the regions overlap? Independent is a question about influence — does knowing one change the other?
• 5. Treating an experimental result as a certainty A relative frequency is an estimate. An expected frequency is a prediction from a model. Neither is a promise. "The expected number of sixes in 300 rolls is 50, so there will be 50 sixes" is wrong, and so is "the spinner landed on blue 84 times out of 240, so \(P(\text{blue})=0.35\)" stated as a fact rather than as an estimate. Fix Use the words the syllabus uses: estimate for a relative frequency, expected for a model prediction. Write \(\approx\), not \(=\), when the number came out of an experiment.

• The one boundary worth memorising. If a question makes you take something out and not put it back, it is an Extended question. Every without-replacement tree, every changing denominator and every conditional probability in this chapter sits inside an Extended block for exactly that reason.
• What the syllabus updates changed here. Version 2, published in February 2024, made exactly one change to Topic 8: the guidance for C8.2.2 and E8.2.2 was updated to include the term random. Version 3, published in May 2024, adjusted page alignment only and is the current version. That single change is why this chapter treats fair, bias and random as three separate ideas with three separate definitions rather than as loose synonyms — see the fair, bias and random clinic inside section 8.2.
• Why the model is worth drawing. Where a multi-part question opens by asking you to complete a Venn diagram or a tree, the later parts are read off what you built. A modelling error in part (a) then propagates through the rest — and, conversely, the completed diagram is itself the communicated method the syllabus asks for. Draw it even when the question does not explicitly ask you to.
• Two phrases worth translating on sight. "At least one" means "one or more" — its complement is "none", which is usually a single route. "Exactly one" means "one and not the other" — on a two-stage tree that is two routes, added. Mistaking one for the other changes the answer, not just the working.
• The complement is a shortcut, not just a definition. Whenever an event is awkward to count directly — "at least one", "not all the same", "more than two" — count the opposite instead and subtract from 1. The opposite is usually a single simple case.
• Three sentences that sound right and are wrong. "It's random, so all the outcomes are equally likely." No — random means unpredictable in advance, not equally likely. A biased spinner is perfectly random. "Each colour came up a different number of times, so the spinner is biased." No — exact equality would be astonishing, not reassuring. Compare with the expected frequency and think about how many trials there were. "I got 7 heads in 10 tosses, so the coin is biased." No — 10 trials tells you almost nothing. Ten thousand would tell you a great deal.
• Do not cancel too early on the non-calculator paper — Paper 1 if you are a Core candidate, Paper 2 if you are Extended. Keeping every route product over the same denominator, here 25, makes adding them trivial and the total-to-1 check instant. Simplify once, at the end.
• Where the marks are. A conditional-probability question is marked on the denominator. Write the restricted total down explicitly and label it — "of the 92 bus travellers", "the 28 who take Art", "the routes with at least one red total \(\frac{9}{10}\)" — before you write the fraction. It communicates the method the syllabus asks you to show, and it makes the answer almost impossible to get wrong.
• The habit all four share. Each solution finishes by checking a set of exhaustive probabilities against 1 — eight outcomes of \(\frac18\), four spinner probabilities, same-colour against different-colour, or the two options the condition leaves open. That check costs one line and catches almost every modelling error this topic can produce.
• Marking yourself honestly. Give yourself the method marks only if the working was on your page before you looked — the completed tree, the filled Venn, the products before they were added. A correct final answer with no visible model would not earn full marks on a question that asks for working.

• Every candidate takes exactly two components, and which two depends on the route. Core candidates take Papers 1 and 3; Extended candidates take Papers 2 and 4. Nobody takes all four, and the paper numbers do not overlap between the routes — so "Paper 2" means something entirely different to a Core and an Extended candidate.
• Both of your papers can ask about every part of Topic 8 that belongs to your route. There is no "probability paper" and no part of the topic that is safe to skip because it belongs to the other component.
• No probability formula is printed on the paper. The list of formulas on page 2 — and there is a separate list for Core and for Extended — covers areas, volumes and surface areas only. Neither list contains anything for probability, so the complement rule, and at Extended the addition rule \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), have to be known.
• Every probability lies between 0 and 1. An answer outside that range is certainly wrong, and it is the fastest check you own.
• Answers should be given in their simplest form unless the question says otherwise, so \(\frac{6}{20}\) should be written \(\frac{3}{10}\).
• Do not mix fractions and decimals inside one number. The syllabus's mathematical conventions state that a combination such as \(\frac{0.3}{4}\) is not acceptable as a final answer.