Build the sample space before calculating

The idea

A fraction is a model: its denominator says what may be selected, and its numerator says which of those possibilities count as a success. Before calculating, finish the sentence “the selection is made from …”. If a question says “given that” or “from those who”, the eligible group has been restricted. A percentage taken within that group can differ from a percentage taken within the whole population, even when the numerator is unchanged. A verbal description can also define the event indirectly. “At least one success” includes exactly one, two, three and any larger permitted number of successes. Instead of counting those cases separately, consider whether the opposite event is easier. With independent trials, the probability of no successes is a product of failure probabilities. Subtract that product from 1 only after checking that the trials really are independent. Reported data require equal care. A rounded mean represents an interval of possible true means, not necessarily one exact value. A uniform shift in a list changes its mean but leaves all deviations from the mean unchanged. These are different kinds of information: a total, a centre, a spread and a rounded display are not interchangeable. The right model tells you which calculations are justified.

A method to try

1. Name the eligible group and the success group in words. For a two-way table, mark the relevant row or column total before dividing. Keep the condition in the denominator; do not reverse “A among B” into “B among A”.
2. Translate the event precisely. For “at least one”, try the complement “none”. For overlapping groups, count the union once by subtracting the overlap, or recover the overlap from the two group totals and the union.
3. Preserve how the data were recorded. Convert rounding to lower and upper bounds, noting whether each endpoint is included. For a transformed list, describe how each value and the mean move before deciding what happens to spread.
4. Check the result against the model. A probability must lie from 0 to 1; its denominator must count each eligible outcome once. Test a claimed extreme total by reconstructing a possible list and checking that its mean rounds as required.

Make it your own

Before opening the companion practice, explain aloud why “completed among morning bookings” and “morning among completed bookings” use different denominators. After an error, identify whether it came from the sample space, the event, an unjustified independence assumption or a rounding boundary. Redo the model in words before repeating the arithmetic.

This lesson is one part of a short starter pathway. It does not cover the full exam syllabus or establish a predicted score.