Historical Context & Motivation
Humans have been gambling for thousands of years, but for most of that time nobody had a systematic way to calculate their chances of winning. Dice carved from animal bones have been found in archaeological sites dating back over 5,000 years, yet the mathematical study of probability only began in the 17th century. The spark came from a simple question about a dice game, and the ideas that followed now underpin everything from weather forecasting to medical testing to machine learning.
The central question probability answers is straightforward: how likely is a particular outcome? To answer it, we need a clear language for describing all possible outcomes, the specific outcomes we care about, and the tools — like Venn diagrams and tree diagrams — that help us organize and calculate. That is exactly what IB SL 4.3 covers.
Core Principles & Definitions
Before you can calculate any probability, you need to understand a handful of foundational ideas. These concepts form the vocabulary of the entire topic, and they appear in virtually every probability question you will encounter on the IB exam.
Experiment & Outcome
Sample Space (U)
Event (A, B, …)
Complementary Event (A′)
Equally Likely Outcomes
Venn Diagrams — Visualizing Events
A Venn diagram uses overlapping circles inside a rectangle to show how events relate to each other and to the sample space. The rectangle represents U, and each circle represents an event. The region where circles overlap represents outcomes belonging to both events simultaneously — the intersection A ∩ B. The total area covered by both circles represents outcomes in at least one event — the union A ∪ B.
The Venn diagram makes the addition rule visually obvious: if you add P(A) + P(B), you count the overlap region twice, so you must subtract it once. That gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). You can literally see this on the diagram — try shading A, then shading B, and notice the intersection gets double-shaded.
Mathematical Framework
The equations below are the core formulas you need for SL 4.3. Each one translates a visual idea — from Venn diagrams or counting — into a precise calculation.
Tree Diagrams — Sequential Events
While Venn diagrams shine when you have two overlapping events, tree diagrams are the go-to tool when events happen in sequence — one after the other. Each branch represents a possible outcome at a given stage, and you write the probability of that outcome along the branch. To find the probability of a specific path from start to finish, you multiply along the branches. To find the probability of multiple paths that all satisfy the same condition, you add across the paths.
Notice how the second-stage probabilities change because the marble is not replaced. After drawing a red marble first, only 2 red and 2 blue remain, so the probability of red on the second draw becomes 2/4 rather than 3/5. This is what "without replacement" means in practice, and tree diagrams make it easy to track these changing probabilities stage by stage.
- Multiply along a path to get the probability of that specific sequence of outcomes.
- Add across paths when you want the probability of any one of several sequences (e.g., "at least one red").
- All paths must sum to 1 — this is a great self-check for your work.
Worked Example — Venn Diagram Problem
In a class of 30 students, 18 study Biology (B) and 12 study Chemistry (C). 7 students study both. A student is selected at random. Find the probability that the student studies Biology or Chemistry (or both), and the probability that the student studies neither subject.
Venn Diagrams vs. Tree Diagrams — When to Use Which
Both Venn diagrams and tree diagrams are representations of probability, but they excel in different situations. Knowing which tool to reach for can save you valuable time on the IB exam.
| Feature | Venn Diagram | Tree Diagram |
|---|---|---|
| Best for | Overlapping events from a single experiment | Sequential (multi-stage) experiments |
| Shows intersections | Yes — overlap region is clearly visible | Indirectly — combine branch paths |
| Shows conditional probabilities | Not directly | Yes — second-stage branches are conditional |
| Number of events | Practical for 2 or 3 events | Handles any number of stages |
| Replacement matters? | Not typically shown | Easily captured by changing branch probabilities |
Connection to Conditional Probability & Beyond
The ideas in SL 4.3 are the foundation for everything that comes next in IB probability. Once you are comfortable with sample spaces, events, and the addition rule, you are ready for conditional probability (SL 4.4) and independent events (SL 4.5). In fact, tree diagrams already hint at conditional probability: the second-stage branch probabilities depend on what happened at the first stage.
| Concept | SL 4.3 (This Lesson) | SL 4.4–4.5 (Next Steps) |
|---|---|---|
| Sample space | List or diagram all outcomes | Restricted sample spaces (given conditions) |
| Intersection | A ∩ B from Venn diagrams | P(A ∩ B) = P(A) × P(B | A) |
| Addition rule | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | Extended to conditional scenarios |
| Tree diagrams | Multiply along paths, add across paths | Bayes' theorem applications |
If you are planning to take the HL course, these fundamentals also feed into probability distributions (HL 4.12–4.14), where you model the number of times an event occurs over many trials. For now, focus on building a rock-solid understanding of sample spaces, the addition rule, and diagram skills — everything else builds on these.
Practice Problems
Lesson Summary
Probability measures how likely an event is to occur, always between 0 and 1. The sample space U is the set of all possible outcomes, and an event is any subset of U. For equally likely outcomes, P(A) = n(A) / n(U). The complement rule P(A′) = 1 − P(A) lets you find the probability that an event does not occur, which is often the faster approach.
Venn diagrams visualize overlapping events with circles, making the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) intuitive — subtract the overlap to avoid double-counting. Tree diagrams handle sequential events by multiplying along branches and adding across paths, naturally accommodating with-replacement and without-replacement scenarios. Together, these tools give you a complete toolkit for tackling any probability question at the SL level.