IA idea · Probability & chance
Is the “hot hand” real? Streaks in my free throws
Research question
After making a free throw, is my probability of making the next one significantly higher than after a miss, or are my streaks consistent with independent shots?
Make it yours, then agree it with your teacher.
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Why it makes a good exploration
Psychologists argued for decades about whether the hot hand exists; a 2018 paper found a subtle bias in the original analysis. Testing your own shots — and understanding that bias — is a fascinating probability IA.
The mathematics you'll need
- Conditional probability
- Runs and the expected longest streak under independence
- Simulation to find the distribution of streaks
- Chi-squared test of independence (make/miss × previous shot)
The statistics, step by step
Worked with every number shown, with what examiners look for and the common mistakes: Chi-squared test for independence. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.
Where the data comes from
Shoot and record 300+ free throws in sessions of 50 (or use any repeatable skill: darts, penalty kicks).
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
A possible outline
- Explain independence and what a hot hand would mean.
- Record sequences carefully.
- Compare P(hit | hit) and P(hit | miss).
- Simulate independent sequences to judge whether your streaks are unusual.
- Reflect on the Miller–Sanjurjo bias in small samples.
Pitfalls that cost marks
- Too few shots to detect anything.
- Pooling sessions with different conditions.
- Not understanding why short sequences bias the conditional estimate.
Showing personal engagement
- Use your own sport.
- Record how you felt during streaks and compare with the data.
- Test a teammate's shots too (with consent).
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 | Conditional probability; Runs and the expected longest streak under independence |
| AA HL | Good fit | Conditional probability; Runs and the expected longest streak under independence |
| AI SL | Good fit | Conditional probability; Runs and the expected longest streak under independence |
| AI HL | Good fit | Conditional probability; Runs and the expected longest streak under independence |
Level: Solid. Needs some independent work beyond class examples. 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: too few shots to detect anything; and pooling sessions with different conditions. Say how each one affects your answer.
Use of mathematics (E)
SL: conditional probability; runs and the expected longest streak under independence; simulation to find the distribution of streaks; chi-squared test of independence (make/miss × previous shot), 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
Reproduce the small-sample bias with a simulation and explain it in your own words.
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 Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Conditional probability, runs and the expected longest streak under independence, simulation to find the distribution of streaks and chi-squared test of independence (make/miss × previous shot).
Where can I get data for this IA?
Shoot and record 300+ free throws in sessions of 50 (or use any repeatable skill: darts, penalty kicks). Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “Is the “hot hand” real? Streaks in my free throws” filled in: the question, the maths and the data sources.
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