Basic Probability (AA SL)
A probability starts as a count: how many outcomes make up the event you want, out of all the outcomes that could happen. This page covers reading a probability straight from a sample space and the complement shortcut for "not" questions, with worked examples and the slips that cost marks. It's part of the broader Probability topic.
23 questions on this sub-topic.
Probability from a sample space
Covered under IB syllabus reference SL4.6, which sets out how probabilities are built from counting outcomes, including combined and conditional events - this page focuses only on the single-event starting point.
Basic probability
\(P(A) = \dfrac{n(A)}{n(U)}\)
The number of outcomes in event \(A\) divided by the total number of equally likely outcomes in the sample space \(U\). Not in the formula booklet - this is prior knowledge you're expected to already know.
Complement rule
\(P(A') = 1 - P(A)\)
"Not \(A\)" is everything outside \(A\), so its probability is whatever's left over from 1. Often faster than counting the "not" outcomes directly.
Ready to combine events, or condition on extra information? See Conditional Probability or the full Probability topic page.
Worked examples
A bag has 5 red, 3 blue and 2 green counters. One is drawn at random.
(a) Find \(P(\text{red})\).
(b) Find \(P(\text{not green})\).
Worked solution
(a) P(red). \(P(\text{red})=\dfrac{5}{10}=\dfrac12.\) A1
(b) P(not green). Complement of green: \(1-\dfrac{2}{10}=\dfrac{8}{10}\) M1
\(=\dfrac45.\) A1
Of 200 commuters: 120 take the bus, of whom 90 are late at least once; 80 take the train, of whom 20 are late at least once. A commuter is chosen at random.
(a) Find \(P(\text{late at least once}).\)
(b) Find \(P(\text{bus} \mid \text{late}).\)
Worked solution
Organise the numbers in a two-way table: bus 90 late / 30 not late (120 total); train 20 late / 60 not late (80 total); 200 overall. M1
(a) Late \(= 90 + 20 = 110\), so \(P(\text{late}) = \dfrac{110}{200} = 0.55.\) A1
(b) Restrict attention to the 110 late commuters; 90 of them took the bus: M1
\(P(\text{bus} \mid \text{late}) = \dfrac{90}{110} = \dfrac{9}{11} \approx 0.818.\) A1
Common mistakes
- Dividing by the wrong total. The denominator is the total number of outcomes in the whole sample space, not just one category of it - miscounting the total (e.g. forgetting one group of items exists) throws off every answer built from it.
- Treating "not A" as a fresh count instead of using the complement. It's usually faster and safer to compute \(1-P(A)\) than to recount every outcome that isn't \(A\) from scratch, especially once a table or diagram is already partly built.
- Forgetting probabilities must lie between 0 and 1. A quick sanity check - does the fraction come out sensible, and do the parts of a full breakdown sum to 1 - catches a lot of arithmetic slips before they cost marks.
Ready to practise properly?
23 basic-probability questions, marked instantly like the real exam.
Quick answers
How do you calculate a basic probability?
\(P(A) = \dfrac{n(A)}{n(U)}\), the number of outcomes in event \(A\) divided by the total number of equally likely outcomes in the sample space \(U\).
What is the complement rule in probability?
\(P(A') = 1 - P(A)\), where \(A'\) is the event "\(A\) does not happen". Often quicker than counting the "not \(A\)" outcomes directly. See the full GDC guidance on the Probability topic page for evaluating expressions like this on your calculator.