Favourable outcomes over total outcomes: When outcomes are equally likely, the probability of event A is the number of favourable outcomes ÷ the total number of outcomes.
Every probability lies between 0 and 1.
The sample space of two dice (36 outcomes): count the favourable squares over 36 to get the probability.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
IB-style question — single event
A bag holds 5 red, 3 blue and 2 green counters.
One is taken at random.
Find the probability it is red.
Step by step
- Total counters.
- Favourable ÷ total.
Final answer
P(red) = 1/2.
Probabilities never exceed 1: If you get a probability above 1 or below 0, you've made an error — recheck your counts.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
P(not A) = 1 − P(A): The complement of A is 'A does not happen', with P(A′) = 1 − P(A).
This is the fast route to 'at least one' questions: P(at least one) = 1 − P(none).
IB-style question — at least one
Two fair dice are rolled.
Find the probability of getting at least one six.
Step by step
- P(no six on one die) = 5/6, so P(no six on both).
- Complement.
Final answer
P(at least one six) = 11/36.
'At least one' → do the opposite: Counting every 'at least one' case is slow; 1 − P(none) is almost always faster.
Stop wasting time on topics you know
Our AI identifies your weak areas and focuses your study time where it matters. No more overstudying easy topics.
List every equally likely outcome in a grid: A sample space diagram (grid or list) shows every equally likely outcome.
For two dice there are 36 outcomes; count the ones matching the event and divide by 36.
IB-style question — two dice
Two fair dice are rolled and the scores added.
Find the probability the total is 7.
Step by step
- Total outcomes in the grid.
- Outcomes giving 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1).
Final answer
P(total = 7) = 1/6.
Order matters in the grid: (2,5) and (5,2) are different cells — count ordered outcomes so the total is 36.
Multiply along the chain — and update the totals: For events one after another, multiply the probabilities along the way.
Without replacement, the counts change after each draw (one fewer item, and one fewer of that type).
IB-style question — without replacement
A box has 4 red and 6 blue pens.
Two are drawn without replacement.
Find the probability both are red.
Step by step
- First red: 4 of 10.
- Second red: now 3 of 9. Multiply.
Final answer
P(both red) = 2/15.
Reduce the totals each time: After the first draw there are 9 pens left, not 10 — forgetting to reduce both totals is the classic slip.