An event is a set, not a number: building the algebra of events before the probability measure in a school progression
Vincent Tariel-Adam · arXiv:2610.34803 [math.HO] · 27 October 2026 · CC BY 4.0
Abstract
Asked what an event is, pupils often answer with a fraction. This paper argues that school probability should separate two learnings and order them: the algebra of events (random experiment, outcomes, sample space, events as sets of outcomes, the operations not, or, and) and the probability measure, which attaches a number to each event. Before learning to compute the probability of an event, pupils should learn to build, identify and handle the event. The thesis rests on the structure of finite probability, in which the rules are facts about events carried over by an additive function, and on the history of the subject, in which errors such as d'Alembert's two tosses concerned the weighting of outcomes and were resolved by refining events into equally likely outcomes. Five design principles are stated: two objects with two names; the event before its probability; modal judgements (certain, possible, impossible) on sets before numbers; the algebra of events taught as such, with the rules derived from it; the sample space as a modelling choice, with frequencies last, as statistical estimates, and the separation repeated for conditional probability and random variables. They are confronted with the probability chapters, from age 9 to 18, of the French-curriculum courses published at https://commeunjeu.org, with research and other curricula. In primary school events precede probabilities without set notation, which comes at age 12; the lessons apply the principles and prove the rules from events in lower secondary school. But, in the version analysed, events account for 18 to 44 percent of the exercises, proofs are hidden in upper secondary school, and automatically corrected calculations record only the number. It proposes number-free tasks on events for ages 6 to 8 and changes to the International Baccalaureate curriculum. No empirical evaluation is reported; hypotheses are stated.
Keywords: probability; events; sample space; algebra of events; modelling; didactics of mathematics; curriculum design
For the classroom
The guide How to Teach Probability takes from the article what is useful in class: what to show, which exercises to give, which errors to watch for.
The chapters studied
The article examines these IB chapters and proposes changes to them:
- PYP 0: Sorting, Picture Graphs and Chance
- PYP 1: Chance
- PYP 2: Chance
- PYP 3: Chance
- PYP 4: Probability
- PYP 5: Probability
- MYP 1: Probability
- MYP 2: Probability
- MYP 3: Sets and Venn Diagrams
- MYP 3: Probability
- MYP 4: Probability
- MYP 5: Probability
- MYP 5: Conditional Probability
- DP 1: Sets and Venn Diagrams
- DP 1: Probability
- DP 2: Discrete Random Variables
How to cite
Tariel-Adam, V. (2026). An event is a set, not a number: building the algebra of events before the probability measure in a school progression. arXiv:2610.34803 [math.HO]. https://doi.org/10.48550/arXiv.2610.34803
@misc{tarieladam2026events,
author = {Tariel-Adam, Vincent},
title = {An event is a set, not a number: building the algebra of events before the probability measure in a school progression},
year = {2026},
eprint = {2610.34803},
archivePrefix = {arXiv},
primaryClass = {math.HO},
doi = {10.48550/arXiv.2610.34803},
url = {https://arxiv.org/abs/2610.34803}
}