The basics that everything rests on
Probability runs from 0, impossible, to 1, certain, and is written as a fraction, decimal or percentage — never as a ratio or in words like "2 in 5 chance" when a value is asked for.
For equally likely outcomes, the probability is the number of favourable outcomes divided by the total number of outcomes.
Two properties do most of the work:
• An exhaustive set of outcomes covers everything that can happen, so the probabilities sum to 1 • Mutually exclusive events cannot happen together, so their probabilities can be added
Together these give the rule used constantly: P(not A) = 1 − P(A). When a question asks for the probability of at least one of something, working out the probability of none and subtracting from 1 is nearly always faster.
Expected frequency is probability multiplied by the number of trials. If a biased spinner lands on red with probability 0.3, then in 200 spins you would expect 0.3 × 200 = 60 reds. It is an expectation rather than a guarantee, and saying so is sometimes worth a mark.
Relative frequency is the other direction — it estimates probability from results actually observed, as the number of times something happened divided by the number of trials. The more trials, the closer relative frequency tends to get to the theoretical probability, which is how you judge whether a coin or dice is biased.
- Probability
- a measure of how likely an outcome is, running from 0 (impossible) to 1 (certain), written as a fraction, decimal or percentage
Listing outcomes systematically
A possibility space lists every outcome of a combined experiment, usually as a grid. Two dice give a 6 by 6 grid with 36 outcomes, from which any probability can be counted directly — and it is far more reliable than trying to reason about the totals.
Frequency trees record what actually happened rather than probabilities: the numbers on the branches are counts, and they must add up correctly at every stage.
A two-way table classifies data by two characteristics at once, with the totals along the edges. Fill in every cell you can from the totals before answering — most questions become one subtraction once the table is complete.
- Possibility space
- a list or grid of every outcome of a combined experiment, from which any probability can be counted directly
Independent and dependent events
Independent events do not affect each other — rolling a dice twice, or taking a counter and putting it back. The probabilities on the second set of branches are the same as the first.
Dependent events do affect each other, which in practice means taking something without replacement. One item has gone, so both the numerator and the denominator change on the second set of branches — and that change is where nearly all the marks in this topic sit.
On a tree diagram: multiply along the branches to find the probability of a particular path, and add between the paths when more than one satisfies the question. Each pair of branches sums to 1, which is a quick check that the tree is right before any calculation.
The underlying assumption the specification asks about is exactly this: multiplying probabilities assumes the events are independent. If they are not, the second probability has to be conditional on the first — which is the next section.
Worked example
A bag holds 5 red and 3 green counters. Two are taken without replacement. Find the probability that they are the same colour.
- There are 8 counters, so the first pick is 5⁄8 red or 3⁄8 green.
- Without replacement, only 7 counters remain for the second pick.
- P(both red) = 5⁄8 × 4⁄7 = 20⁄56, because after a red is taken 4 reds remain out of 7.
- P(both green) = 3⁄8 × 2⁄7 = 6⁄56.
- Same colour means either path, and they are mutually exclusive, so add them.
- 20⁄56 + 6⁄56 = 26⁄56
Answer: 26⁄56 = 13⁄28. The denominators change from 8 to 7 because a counter has gone — if the question had said with replacement, both would stay 8.
- Independent events
- events that do not affect each other, so the probabilities on the second set of branches are the same as the first
- Dependent events
- events that do affect each other, which in practice means taking something without replacing it
Conditional probabilityHigher tier
Conditional probability is the probability of one event given that another has already happened. The wording to watch for is "given that", or a question that tells you an outcome and then asks about a related one.
The effect is to shrink the set of possibilities you are working within. You are no longer considering everything — only the cases where the given condition holds.
The method is the same whichever diagram you are working from: identify the restricted group the condition creates, and use that as the new denominator.
• From a two-way table — use only the relevant row or column, and its total • From a Venn diagram — count only within the region the condition describes • From a tree diagram — the second set of branches already shows the conditional probabilities, so read them directly
Expected frequencies often make conditional questions easier than fractions do. Turning the probabilities into counts out of 100 or 1000 people, filling in a table, and then reading off the answer avoids most of the arithmetic.
Worked example
In a class of 30, 18 study French and 14 study Spanish; 7 study both. A student who studies French is chosen at random. Find the probability that they also study Spanish.
- The condition is that the student studies French, so the restricted group is the 18 French students.
- That 18 becomes the denominator, not the 30.
- Of those 18, the ones who also study Spanish are the 7 in the overlap.
- So the probability is 7 out of 18.
Answer: 7⁄18. Answering 7⁄30 is the standard error — it ignores the condition, which was the whole point of the question.
Common mistake
"Given that" changes the denominator. If the answer still has the original total on the bottom, the condition has not been applied.
- Conditional probability
- the probability of one event given that another has already happened, signalled by the wording 'given that'
Before you practise
- Probability runs from 0 to 1, and an exhaustive set of outcomes sums to 1.
- P(not A) = 1 − P(A) — and for "at least one", find P(none) and subtract.
- Mutually exclusive events cannot happen together, so their probabilities add.
- Expected frequency = probability × number of trials.
- Relative frequency estimates probability from results, and approaches the theoretical value as trials increase.
- Use a possibility space grid for combined experiments — count, do not reason.
- Complete a two-way table before answering anything from it.
- Multiply along tree branches, add between paths, and check each pair sums to 1.
- Independent events keep the same probabilities; dependent events change both numerator and denominator.
- Multiplying probabilities assumes independence — that is the assumption questions ask about.
- "Given that" restricts the group and changes the denominator.
- Converting to expected frequencies out of 100 makes most conditional questions easier.