Basic Probability (AI SL)
Before you get to Venn diagrams, tree diagrams or conditioning on other events, every probability question comes back to one idea: count the outcomes you want, count the outcomes that are possible, and divide. This page covers reading a sample space or table of outcomes directly, plus the complement rule for "not" questions - the two skills every harder probability question in this topic still leans on. It's part of the broader Probability topic.
38 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL4.6: use of sample space diagrams and tables of outcomes to calculate probabilities, \(P(A)=\dfrac{n(A)}{n(U)}\).
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 - prior knowledge, but everything else on this topic builds on it.
Complement
\(P(A') = 1 - P(A)\)
\(A'\) means "\(A\) does not happen". Useful whenever "not" or "at least one" is easier to find by working out the opposite case first.
Ready to combine two events, or condition one on another? See Probability.
Worked examples
A spinner has 8 equal sectors: 3 red, 2 blue, 3 green.
(a) Find the probability of landing on red.
(b) Find the probability of not landing on blue.
Worked solution
(a) Red. 3 of the 8 sectors are red: \(P(\text{red})=\dfrac{3}{8}.\) A1
(b) Not blue. Use the complement: M1
\(P(\text{not blue})=1-P(\text{blue})=1-\dfrac{2}{8}=\dfrac{6}{8}=\dfrac34.\) A1
Two fair six-sided dice are rolled.
(a) Find the probability the sum is 7.
(b) Find the probability the sum is greater than 9.
Worked solution
(a) Sample space. Two dice give \(6\times6=36\) equally likely outcomes. M1
Sum 7. The pairs \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\) - 6 outcomes: \(\dfrac{6}{36}=\dfrac16.\) A1
(b) Sum \(>9\) (i.e. 10, 11, 12). Sum 10: 3 ways; sum 11: 2 ways; sum 12: 1 way. M1
Total \(3+2+1=6:\) \(\dfrac{6}{36}=\dfrac16.\) A1
Common mistakes
- Listing outcomes that aren't equally likely. \(P(A)=\dfrac{n(A)}{n(U)}\) only works if every outcome in the sample space has the same chance - "sum of two dice" has 11 possible totals, but they are not equally likely, so you must count the 36 underlying pairs instead.
- Forgetting the complement flips the count, not just the label. \(P(A')=1-P(A)\) needs the correct \(P(A)\) first - a wrong probability for the event still gives a wrong "not" probability, just relabelled.
- Leaving a probability as a raw fraction that doesn't simplify or convert cleanly. Always simplify fractions and check the decimal makes sense (between 0 and 1) before moving on to the next part of a question.
Ready to practise properly?
38 basic-probability questions, marked instantly like the real exam.
Quick answers
How do I find a basic probability from a sample space?
Count the outcomes that satisfy your event, then divide by the total number of equally likely outcomes in the sample space: \(P(A) = \dfrac{n(A)}{n(U)}\). Listing the full sample space first avoids missing or double-counting outcomes.
What's the complement rule and when do I use it?
The complement rule says \(P(A') = 1 - P(A)\), where \(A'\) means "\(A\) does not happen". It's fastest whenever a question asks for "not" or "at least one" and the opposite case is easier to count directly.