IB Maths AA SLTopic 4 — ProbabilityPaper 1 & 2~10 min read
Probability & Types of Events
Probability is the maths of chance — what’s the likelihood that something will happen? This note covers all the basic vocabulary and rules: outcomes, events, the probability formula, and how to combine events using AND and OR.
📘 What you need to know
Probability formula: P(A) = n(A)n(U) = favourable outcomestotal outcomes
Probability is always between 0 and 1: 0 means impossible, 1 means certain.
Complement: P(A‘) = 1 − P(A) — “not A“.
Union (OR): P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Intersection (AND): P(A ∩ B) = P(A) × P(B|A).
Expected occurrences in n trials: np (where p is the probability).
The vocabulary you’ll meet
Probability has a few specific words you need to know. They sound technical, but they’re really just everyday concepts:
Experiment
Any repeatable activity with a result you can record.
e.g. rolling a dice, flipping a coin
Trial
One single repeat of the experiment.
e.g. one roll of the dice
Outcome
A possible result of one trial.
e.g. rolling a 4
Event
An outcome OR a collection of outcomes. Usually denoted by capital letters (A, B).
e.g. “rolling an odd number” = {1, 3, 5}
Sample space (U)
The set of all possible outcomes of an experiment.
e.g. {1, 2, 3, 4, 5, 6} for a dice
n(A) and n(U)
n(A) = number of outcomes in event A. n(U) = total number of outcomes.
e.g. for “odd dice”: n(A) = 3, n(U) = 6
The basic probability formula
If every outcome is equally likely (a fair dice, an unbiased coin), the probability of an event is just:
This formula is in your formula booklet. As long as the outcomes are equally likely, it always works. To use it: count how many outcomes count as the event you care about, count the total, divide.
The probability scale
All probabilities live between 0 and 1
0
0.25
0.5
0.75
1
impossible
even chance
certain
If you ever calculate a probability above 1 or below 0, you’ve made a mistake!
Theoretical vs experimental probability
Theoretical probability = calculated using the formula. Works when outcomes are equally likely.
Experimental probability (or relative frequency) = based on actual results from trials. = frequency of outcometotal trials
For a fair coin, theoretical P(heads) = 0.5. But if you flip it 10 times and get 7 heads, the experimental probability is 0.7. Run more and more trials and the experimental probability gets closer to 0.5 — that’s the “law of large numbers”.
Expected number of occurrences
If the probability of an outcome is p, and you repeat the experiment n times, you expect it to occur:
Expected occurrences
Expected number = n × p
Example: If you roll a dice 60 times, the expected number of 6’s is 60 × 16 = 10.
📍
“Expected” doesn’t mean “guaranteed”
If you roll a dice 60 times you might get 8 sixes, or 13, or 7 — not exactly 10. But on average, over many sets of 60 rolls, you’d expect 10 sixes. The expected value is a long-run average, not a promise.
The complement of an event
The complement of event A is “everything exceptA” — the event that A doesn’t happen. We write it as A‘.
The complement formula
P(A) + P(A‘) = 1 ⟹ P(A‘) = 1 − P(A)
Example: If P(“rolling a 6”) = 16, then P(“not rolling a 6”) = 1 − 16 = 56.
🧠
Memory trick: “1 minus the easy one”
If finding “A” is hard but finding “not A” is easy, calculate the easy one and subtract from 1. This is a powerful exam shortcut — especially for “at least one” questions.
Combining events — AND, OR, NOT
When you have two events A and B, you can combine them in different ways:
A’ — complement
A‘
“NOT A“ — event A does not happen.
e.g. NOT rolling a 6 → {1,2,3,4,5}
A ∩ B — intersection
A ∩ B
“A AND B“ — both events happen.
e.g. rolling an even number AND a multiple of 3 → just {6}
A ∪ B — union
A ∪ B
“A OR B (or both)” — at least one happens.
e.g. rolling an even number OR a multiple of 3 → {2,3,4,6}
A | B — conditional
A | B
“A GIVEN B“ — probability of A happening given that B already happened.
covered in detail in the conditional probability note
🧠
Memory trick: “U for Union, like a U-shape holding both”
The union symbol ∪ looks like a cup — it holds both circles. The intersection symbol ∩ looks like a bridge — it’s only the bit where they overlap. AND = overlap = ∩. OR (or both) = full cup = ∪.
Formulas for combined events
The OR formula (union)
Probability of A or B (or both)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
🤔 Why subtract P(A ∩ B)?
If you just added P(A) + P(B), you’d be counting the overlap twice — once in P(A) and once in P(B). Subtracting P(A ∩ B) once removes the duplicate.
Imagine 60% of students play football and 30% play tennis. If 20% play both, then the percentage who play at least one is 60 + 30 − 20 = 70% (not 90%).
The AND formula (intersection)
Probability of A and B both happening
P(A ∩ B) = P(A) × P(B | A)
In words: “probability of A, multiplied by the probability of B happening given that A has happened”. The conditional bit P(B|A) handles the case where the events affect each other (covered fully in the conditional probability note).
📍
Quick test for spotting which formula to use
Look at the wording: “A AND B” or “both” or “and then” → use the intersection formula (P(A) × P(B|A)). “A OR B” or “at least one” or “either” → use the union formula.
Worked examples
WE 1
List outcomes and find probabilities
Dave has two fair spinners. Spinner A has three sides numbered 1, 4, 9 and Spinner B has four sides numbered 2, 3, 5, 7. Dave spins both and forms a two-digit number using A for the first digit and B for the second digit.
Let T be the event that the two-digit number is a multiple of 3.
(a) List all possible two-digit numbers. (b) Find P(T). (c) Find P(T‘).
Use a 2-way table to list all outcomes systematically.part (a) — list outcomesFirst digit (A): 1, 4, 9. Second digit (B): 2, 3, 5, 7.3 × 4 = 12 possible two-digit numbers.Outcomes: 12, 13, 15, 17, 42, 43, 45, 47, 92, 93, 95, 97part (b) — find p(t)Find which numbers are multiples of 3:12, 15, 42, 45, 93 → 5 multiples of 3Use formula:P(T) = n(T)n(U) = 512P(T) = 512part (c) — find p(t’)Use complement:P(T’) = 1 − P(T) = 1 − 512P(T’) = 712a 2-way table is the safest way to list all outcomes — fewer mistakes!
WE 2
Use the OR formula
In a class, P(plays football) = 0.6, P(plays tennis) = 0.3, and P(plays both) = 0.2. Find the probability that a student picked at random plays football or tennis (or both).
“OR (or both)” → use the union formula.Use:P(F ∪ T) = P(F) + P(T) − P(F ∩ T)P(F ∪ T) = 0.6 + 0.3 − 0.2P(F ∪ T) = 0.7P(F ∪ T) = 0.7don’t forget to subtract the overlap — otherwise you’d count it twice!
WE 3
Find the expected number of occurrences
A spinner has 5 equal sectors, one of which is red. The spinner is spun 200 times.
(a) Find the probability of getting red on a single spin. (b) Find the expected number of red sectors in 200 spins.
Equally likely sectors → use n(A)/n(U). Expected = np.part (a)1 red out of 5 sectors:P(red) = 15 = 0.2P(red) = 0.2part (b)Use np with n = 200, p = 0.2:200 × 0.2 = 40Expected = 40 reds“expected” is a long-run average — won’t be exact every time
WE 4
Use the complement for “at least one”
A coin is flipped 3 times. Find the probability of getting at least one head.
“At least one head” is hard to count directly — but its complement (no heads at all) is easy.Complement:“no heads” = “all tails”P(all 3 tails):0.5 × 0.5 × 0.5 = 18Use complement:P(at least 1 head) = 1 − 18 = 78P(at least 1 head) = 78“at least one” is almost always solved by the complement trick
WE 5
Use the AND formula
A bag contains 5 red balls and 7 blue balls. Two balls are drawn without replacement. Find the probability that both balls are red.
“Both red” = R₁ AND R₂ → use the intersection formula.P(first red):512After drawing 1 red, 4 reds left out of 11 balls.P(second red | first red):411Multiply:P(both red) = 512 × 411 = 20132 = 533P(both red) = 533“without replacement” → second probability changes — that’s why we use P(B|A)
💡 Top tips
Always check your answer is between 0 and 1. If it’s not, you’ve made an error somewhere.
Use a 2-way table or list to be systematic when listing outcomes — it stops you missing any.
For “at least one” questions, use the complement: P(at least one) = 1 − P(none).
For “AND” questions, multiply the probabilities (using the conditional probability if events affect each other).
For “OR” questions, add but subtract the overlap so you don’t count it twice.
Read carefully — “with” or “without replacement” changes everything in multi-step problems.
If a question describes outcomes that aren’t equally likely, the basic formula doesn’t work — use the situation-specific information instead.
Keep fractions for accuracy when possible — switch to decimals only at the end.
⚠ Common mistakes
Forgetting to subtract the overlap in the union formula. P(A ∪ B) = P(A) + P(B) − P(A ∩ B) — the minus is essential!
Adding probabilities for “AND” questions. AND means multiply, not add.
Multiplying for “OR” questions. OR means add (and subtract the overlap).
Treating “without replacement” as “with replacement”. The probabilities change after each draw.
Confusing the symbols ∪ and ∩. ∪ is union (OR), ∩ is intersection (AND). Mix them up and the maths goes the wrong way.
Reporting probabilities outside [0, 1]. A probability can never be negative or above 1.
Forgetting the complement shortcut. “At least one” problems are almost always faster as 1 − P(none).
Confusing experimental and theoretical probability. Theoretical comes from the formula; experimental comes from real data. Don’t confuse them in answers.
You’ve got the foundations now. The next note covers independent and mutually exclusive events — two special types of events that simplify the formulas above. After that, things get even more interesting with conditional probability, Venn diagrams, and tree diagrams.
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