How to study

Probability

Where the likelihood chain pays out: from the 0-to-1 scale to conditional probability, tree diagrams, and expected value.

Topic Importance

Probability is where the likelihood chain pays out. It has appeared in 30 questions over the last 12 years — every single year, without fail — and as the Paper Two overview explained, this count also includes the binomial and permutations-and-combinations content that the past papers file under the probability label.

Probability appeared in 30 questions over the last 12 years — and the binomial and counting marks are collected here too

That makes it one of the most reliable investments on Paper Two: a guaranteed question type, a settled set of techniques, and every earlier link of the chain — binomial, then permutations and combinations — cashing in through it.

What the Topic Covers

The topic starts with the concept: what a probability even means. Every probability lives between 0 and 1 — impossible to certain — and connects directly to percentage likelihood.

The probability scale from 0 to 1: zero means impossible, a half means even chance, one means certain

From there the machinery builds: complement probability (something happening and not happening must add to 1), calculating basic probabilities as favourable outcomes over total outcomes, and the "at least one" pattern for repeated events. Then combinations of events: unions (the or-case, minus the double count), mutually exclusive events and conditional probability — the probability of A given B.

Then the visual workhorse: tree diagrams, which lay out sequential events and multiply the probabilities along each branch.

A tree diagram for two flips of a rigged coin with a 60 percent chance of heads: multiplying along the branches gives the probability of each sequence

The topic closes with the applied block: relative frequency and predictions (how many flips of a rigged coin to expect a given number of heads), trials — where the binomial from earlier in the chain returns to handle a series of repeated trials — and expected value, combining outcomes' values with their probabilities.

Example question

A die is rolled 5 times. What is the probability that at least one roll shows a two?

Example question

A rigged coin has a 60% chance of landing heads. How many flips would you expect to need to get 150 heads?

Exam Correlations

Probability appears 30 times over 12 years, and its only substantial partner is statistics, with 6 shared questions. The rest of the table is single co-occurrences.

Questions shared with probability over 12 years: statistics 6, and single questions shared with algebra 1, geometry of the line, axioms and theorems, and trigonometry

The statistics number is exactly the bundling described in the Paper Two overview: one question with two parts, one part probability and one part statistics, sometimes sharing the same data — not a deep overlap in ideas. And remember what this table does not show: the binomial and permutations-and-combinations content has no rows of its own, because it lives inside these 30 probability questions. Algebra 1 appears here as itself rather than inside an "Algebra" bundle.

A probability question has appeared every single year for 12 years. Between the guaranteed appearance and the chain of topics it collects marks for, this is the likelihood block's best return on study time.

Concept Connections

Probability is the summit of the likelihood chain: permutations and combinations feeds in, and nothing serious builds on top.

Permutations and combinations feeds into probability, the summit of the likelihood chain, which nothing serious builds on

Comes from The idea it gives you Where you will use it in Probability

Permutations and Combinations \({}^{15}C_{2},\; \text{total} - \text{none}\) Counting outcomes — favourable over total needs both counted, and complement counting returns as complement probability and the "at least one" pattern.

Binomial \(\dbinom{3}{2}\left(\tfrac{1}{2}\right)^{2}\tfrac{1}{2}\) Repeated trials — the trials sub-topic runs on the binomial machinery: the coefficient counts the ways, the powers carry the odds.

And outward? Very little. Probability might lend statistics a small hand, but not much — the 6 shared questions are exam bundling, not dependency. Once probability is done, the likelihood side of Paper Two is effectively complete, with statistics as its own final piece.

Study Order

There are nine sub-topics in Probability: the concept, the core calculating machinery, and then the applied block. Click each step to see how they build.

Introduction to Probability

What a probability means: a number between 0 and 1, where 0 means impossible and 1 means certain, connecting directly to percentage likelihood.

\(0 \leq P(E) \leq 1\)

Complement Probability

Something either happens or it does not — combined, that is certain. So the probability of an event and the probability of its complement add to 1, and you can find either by subtracting the other.

\(P(\text{not } A) = 1 - P(A)\)

Calculating a Basic Probability

When all outcomes are equally likely, a probability is the number of acceptable outcomes divided by the total number of outcomes — both counted with the toolkit from Permutations and Combinations. The "at least one" pattern combines this with steps 1 and 2: find the chance of never getting it, raise it to the power of the repeats, and subtract from 1.

\(P(\text{at least one two in 5 rolls}) = 1 - \left(\tfrac{5}{6}\right)^{5}\)

Unions

The or-case: the probability of A or B is the sum of the two, minus the probability of both — because the overlap would otherwise be counted twice.

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

Mutually Exclusive Events and Conditional Probability

The probability of A given that B happened, built from the both-happening idea of step 4: the odds of both, divided by the odds of the given event. Mutually exclusive events are the special case where both happening is impossible — so the conditional probability is zero.

\(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)

Tree Diagrams

The visual representation of sequential events: each branch carries a probability, and multiplying along a branch gives the probability of that sequence of events. This is the multiplication of independent events from step 3, drawn out.

\(P(HH) = 0.6 \times 0.6 = 0.36\)

Relative Frequency and Predictions

Probability run forwards: if you know the odds, you can predict the long run. A rigged coin with a 60% chance of heads needs 250 flips to expect 150 heads. And run backwards, observed frequencies estimate unknown odds.

\(n \times 0.6 = 150 \;\Rightarrow\; n = 250 \text{ flips}\)

Trials

The outcome of a series of trials — where the chain connects back to Binomial and Permutations and Combinations. The binomial machinery gives the probability of exactly so many successes across repeated attempts.

\(P(k \text{ successes in } n) = \dbinom{n}{k}\, p^{k} \left(1-p\right)^{n-k}\)

Expected Value

The closing idea: combine each outcome's value with its probability, and you get the value to expect over time — the concept behind fair games, insurance, and a reliable final part of exam questions.

\(E = \displaystyle\sum x_{i}\, p_{i}\)

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