Maths IA · Statistics and probability · topic 20 of 40

Hot streaks: is a player streaky or just luck? · IB Maths IA idea: guide, sample graph and step-by-step plan

great for sport fans, but stay sceptical

Commentators say a player is "on fire". You're going to ask whether that is real or just what randomness looks like, and let a p-value settle the argument.

Difficulty●●○ Medium
KitShot-by-shot data, spreadsheet or GDC
TimeAbout 12 to 15 hours
FitsAI, SL and HL
  • ✓ you follow a sport with shot-by-shot data
  • ✓ you like arguing with a commentator
  • ✓ you can set up a hypothesis test

The big idea

If every shot is an independent chance with the same probability p, long runs of hits still happen by luck. A streaky player would show more clustering than chance allows. You build the independent-shots model, work out how many runs (or how long a streak) chance produces, then test your player's real sequence against it. The maths is binomial probability plus a hypothesis test; runs tests and streak formulas are beyond both courses, so introduce whichever you use. The interest is in the decisions: what counts as a streak, what p is, and whether independence is even fair. Mind the multiple-testing trap if you test many players.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

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Compare several players

Run the same test on 3 to 5 players of different styles or positions.

→ You now compare p-values across players, so you must discuss testing many times and how likely one 'significant' result is by chance alone.
Condition on the last shot

Estimate P(hit after a hit) and P(hit after a miss) separately and compare them.

→ Your model becomes a two-state chain instead of a single binomial; comparing them needs a two-proportion test (beyond the course) or a simulation.
Use a simulation

Simulate thousands of independent sequences with the same p and count how often the longest streak matches or beats the real one.

→ You get an empirical p-value to set against the formula, and a comparison of two methods.
Change the sport

Free throws, penalties, a batter's hits, or a computer-game win streak.

→ The context changes how fair the independence assumption is, which is the heart of your evaluation.
Vary the streak definition

Test runs of 3, 4, 5 and so on, or the longest run, or the total number of runs.

→ You discover that the conclusion can depend on the statistic you chose, a strong point for reflection.
Go further for HL

Derive the mean and variance of the number of runs, or the exact probability of a streak by recursion, rather than quoting a formula.

→ This gives the sophistication and rigour that HL criterion E asks for; a routine test alone will not.

Your Maths IA research question: run the check

say what 'streaky' means FIRST

Write your own question first. Then make sure it ticks every box:

  • Names one player (or a small set) and one clear measure of success per attempt.
  • Says what 'streaky' means in your question: longest run, number of runs, or hits after hits.
  • Says how you will decide, for example a hypothesis test at a stated significance level, and against which model.
  • Makes clear the shots are separate attempts so independence is arguable.
  • Is narrow enough to finish: one season or one competition, with a stated number of attempts.

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The model

Each attempt is a hit or a miss. Under the null model they are independent with one probability p. The test asks how extreme your real sequence is under that model.

nnumber of runs R● hit ○ missa run of 5 hits in a row: how surprising is it?if shots are independent,R follows this curveobserved R:in the tail?check independence BEFORE the test

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why this player or sport; what 'streaky' means to you; what you predict before looking at the data.
Data and cleaning~1.5 pages
Where the shot-by-shot data came from, which games or seasons and why, what you excluded, and the order the shots were taken in.
The independence model~2 pages
Define p, n, the random variable, and the null hypothesis in words and symbols. State what independent and identical mean for shots in a real match.
Streak or runs probability~2 pages
How you work out the chance of a long streak or a given number of runs: formula or recursion, with one worked case and a table of values.
The hypothesis test~2 pages
Hypotheses, test statistic, significance level, p-value and decision, each justified. Check conditions before you use any approximation.
Simulation check~1.5 pages
A simulation of independent sequences to compare with the formula, with the number of trials stated and a summary table or graph.
Does independence hold?~1.5 pages
Evidence for or against it: shots after hits versus after misses, fatigue, defence, changing p over a game.
Conclusion~1 page
What the p-value does and does not say; a plain-English answer to your aim.
Reflection and limits~1.5 pages
Assumptions, the multiple-testing issue, what a non-significant result does not prove, and one next step with a reason. Reflect after each stage too, not only here.
Bibliographynot counted
Data source, any formula sources, software.

What your Maths IA graph could look like

show your significance level on it

Made-up simulation results, to show the form of a strong graph. Yours will use your own p and n. example only

12345678910110.00.10.20.30.40.50.60.70.80.91.0Streak length k (hits in a row)P(longest streak is shorter than k)approx. model, p = 0.5, n = 100approx. model, p = 0.4, n = 100simulated proportion (made up)how long a streak israre enough for YOURsignificance level?ILLUSTRATIVE · MADE-UP DATA
  • Axes say exactly what is plotted: the chance that the longest streak is shorter than k. One minus it is the chance of a streak at least that long.
  • A simulated cloud sits close to the model curve, which is evidence your formula (or approximation) is sound.
  • Two curves for two values of p show how much the answer depends on the success rate.
  • Mark 1 − α as a horizontal line (0.95 for a 5% level): where the curve is above it, a streak of length k is rare enough to call surprising.
  • Another graph to consider: a histogram of the number of runs from your simulation, with your player's observed value marked.

Your game plan

trial run by hand, then software

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Pick a measure and a sourceChoose a sport where each attempt is separate (free throws, penalties, serves). Find shot-by-shot data in order. Write down your prediction first.
  2. Read the maths you needBinomial distribution, the idea of a null hypothesis, p-values. Work out what a streak or a run is in a short sequence you can count by hand.
  3. Trial runTake 30 to 50 shots by hand. Count runs, find the longest streak and the proportion of hits. Does your definition of a streak survive contact with real data?
  4. Build the independence modelDefine p and state the assumptions. Decide how p is estimated and whether that estimate affects your test.
  5. Work out the probabilitiesGet the probability of a streak or of a number of runs by a method you can explain. Check it on a tiny case by listing every sequence.
  6. Run the testHypotheses, significance level, test statistic, p-value, decision. Check the conditions the test needs, and say if you had to approximate.
  7. Simulate to checkGenerate many independent sequences, compare with your formula. Where they disagree, find out why.
  8. Question independenceLook at what happens after a hit and after a miss. Is p the same all game? What could make shots depend on each other?
  9. Reflect and extendState the limits of the model and test. Decide on one extension (second player, conditional model, derivation) and say why.
  10. Write, then cutWrite in the layout order. One worked example per method, then tables. Check each symbol is defined once and used consistently, then get your teacher's draft read.

Maths you'll need

Traps to avoid in this Maths IA

  • Treating shots as independent without ever testing that idea.
  • Choosing the streak definition after seeing the data, then reporting only the one that gives a small p-value.
  • Estimating p from the same sequence and ignoring the effect on the test.
  • Testing many players and quoting only the 'significant' one.
  • Reading 'not significant' as 'proved luck'.
  • At HL: running a routine test in the calculator and calling it done; criterion E needs justification, derivation or links to other topics.
  • Software output pasted with no explanation of the statistic.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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