Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

What is the chance of hitting every green light on my route?

AI SLAI HLAA SLAA HL Accessible Also in: Statistics, Modelling

Research question

Using measured cycle times for the traffic lights on my route to school, what is the probability distribution of my total waiting time, and does it match my recorded journeys?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

Your commute is a sequence of random waits. Measuring the lights and modelling the waiting time turns daily frustration into a probability distribution you can test.

The mathematics you'll need

Uniform and mixed discrete-continuous distributions
Expected value of waiting time
Sum of independent random variables (simulation)
Comparing a model with recorded data

Where the data comes from

Time the red and green phases of each light on your route (repeatedly), then record waiting times on 20+ journeys.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Model a single light: P(wait) and expected wait.
Combine lights assuming independence.
Simulate the total waiting distribution.
Compare with recorded journeys.
Reflect: lights are often coordinated (a “green wave”), so independence fails.

Pitfalls that cost marks

Assuming independence without testing it.
Timing phases only once.
Recording journeys at different times of day.

Showing personal engagement

Use your actual route.
Test whether cycling faster or slower changes your total wait.
Investigate a coordinated sequence and model it.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Uniform and mixed discrete-continuous distributions; Expected value of waiting time
AA HL	Fits, but add an HL technique	Uniform and mixed discrete-continuous distributions; Expected value of waiting time
AI SL	Good fit	Uniform and mixed discrete-continuous distributions; Expected value of waiting time
AI HL	Fits, but add an HL technique	Uniform and mixed discrete-continuous distributions; Expected value of waiting time

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: assuming independence without testing it; and timing phases only once. Say how each one affects your answer.

Use of mathematics (E)

SL: uniform and mixed discrete-continuous distributions; expected value of waiting time; sum of independent random variables (simulation); comparing a model with recorded data, each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Model coordinated lights with a fixed offset and find the speed that gives a green wave.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL probability exploration adds.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete probability exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI SL, AI HL, AA SL and AA HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Uniform and mixed discrete-continuous distributions, expected value of waiting time, sum of independent random variables (simulation) and comparing a model with recorded data.

Where can I get data for this IA?

Time the red and green phases of each light on your route (repeatedly), then record waiting times on 20+ journeys. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “What is the chance of hitting every green light on my route?” filled in: the question, the maths and the data sources.

Plan this IA free

Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in

Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)

Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · IA statistics, step by step