Maths IA · Statistics and probability · topic 42 of 97
From rally to match: how small edges become big wins · IB Maths IA idea: guide, sample graph and step-by-step plan
small edge, big win: prove itWin 55% of the rallies and you would expect to win about 55% of the games. You will not. You are going to find out how far off that guess is.
- ✓ you play badminton, table tennis, volleyball or pickleball
- ✓ you like a result that looks wrong until you work it out
- ✓ you want a model you can check by simulation and by real scores
The big idea
A match is built from games, and a game is built from rallies. If you win each rally with the same chance p, the chance of winning a game to 11 is a sum of binomial-type terms, plus a tail for the stage where both players reach 10 points and must win by two. That tail is a geometric series, and it sums to a neat expression. Then a match is built from games the same way. The derivative at p = 0.5 tells you how strongly each stage amplifies a small edge. The formulas are the easy part. Say whether rallies are really independent with the same p, test the model against real scores or a simulation, and compare sports with different scoring rules.
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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Compute the chance of winning a game under both scoring lengths for the same rally chance.
→ A longer game amplifies more. You explain why with a law of large numbers argument and numbers from your formula.Find the chance of winning from 10-10 when you must win by two points.
→ A geometric series, or a short equation from the structure of the problem. You compare both methods.Differentiate the game and match win chances with respect to p at p = 0.5.
→ A single number for how much a small edge is magnified. You compare sports and say which is the most revealing of skill.Simulate thousands of games and matches and compare the results with the formulas.
→ Technology becomes a check on algebra, and you discuss how many runs you need for the error to be small.Use scores from a league or tournament to estimate a rally chance for each player, then predict match results.
→ A real test of the independence assumption. Streaks and serve advantage may make the model fail, and you explain how.Compare traditional and rally scoring, a point cap, or first to 15 against first to 11.
→ New formulas for each rule set, and a question about which rules make the better player win more often.Your Maths IA research question: run the check
which rules? which sport?Write your own question first. Then make sure it ticks every box:
- Names the sport and its scoring rules: points to win, win-by-two, a cap, and the number of games in a match.
- Says what is assumed about rallies (independent, the same chance) and how that will be tested.
- Names what is calculated: game chance, match chance, amplification, or the chance needed for a given result.
- Says how the formula is checked: simulation, real scores, or both.
- Goes beyond a table of results by deriving a formula or comparing two rule sets.
The model
A chain from rally to game to match. Write the rules of your own sport in the boxes and choose a rally chance to follow through.
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.
Your sport and its scoring rules, and a guess: how does a small rally edge show up in the match?
The chance of winning a game as a formula in p, with the sum set up, defined and checked for small cases.
The win-by-two stage solved, by a series or an equation, and added into the game formula.
The match chance built from the game chance, for the number of games in your sport.
Derivatives at p = 0.5, a graph of game and match chance against p, and a comparison of formats.
A simulation that agrees with the formula, and a test of the model against real results.
Back to the aim: how much a small edge is worth, and where the model holds.
Rallies are not identical: serve, fatigue, momentum. Which assumption matters most, and how could you test it?
Sources for the rules and for any scores.
What your Maths IA graph could look like
plot the formula, then the simulationA made-up view to show the form of a strong graph, not the answer: the chance of winning a game to 11 against the chance of winning a rally. Yours will look different. example only
- Both curves on one graph, so the amplification is visible, not just stated.
- Label both axes with what they are. The dotted reference line is the idea 'no amplification'.
- The S-curve here is a shape only; yours comes from your own formula, so plot your formula.
- Simulated points should sit on your curve. Say how many games you simulated for each point.
- Another graph to consider: match chance against rally chance for two different formats.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind the scoring rules of your sport. Before calculating, guess the chance of winning a game if you win 55% of the rallies.
- Fix the rulesWrite down points to win, win-by-two, any cap, and games per match. Say which sport uses which rules.
- Count the waysWork out the chance of winning to 11 with the opponent on 0, 1, ... 9, and add them up. Check small cases by listing.
- Trial run (1 lesson)Simulate 1,000 games in a spreadsheet or Python. Does the result match your first formula? Fix any mismatch now.
- Solve the deuce stageFind the chance of winning from 10-10 and add it into the game formula. Check it against the simulation.
- Build the matchUse the game chance to find the match chance for your format, and graph all three levels against p.
- Test on real scoresEstimate a rally chance from real scores and compare the predicted and the actual game and match results.
- Go further, with a reasonDifferentiate at p = 0.5, or compare two rule sets. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter each result, write one line on what it means for a real player and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every symbol is defined, the simulation agrees with the formula and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Binomial and negative-binomial counting – Count the ways to win 11 points before the opponent reaches 10: for .
- Geometric series – Sum the chances of winning from deuce: , giving .
- Composition of functions – Build the match chance as , with for best of three.
- Differentiation – Find and as amplification factors; use the chain rule for .
- Simulation – Simulate games with a random number generator and compare proportions with the formula, with a stated error.
Key equations
Syllabus links
- 1.3Geometric sequences and series
- 1.8Infinite geometric series
- 1.9Binomial theorem
- 4.5Basic probability and expected frequency
- 4.8Binomial distribution
- 5.6Standard derivatives; chain, product and quotient rules
- 1.3Geometric sequences and series
- 4.5Basic probability and expected frequency
- 4.8Binomial distribution
- 1.11Infinite geometric seriesHL
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
Read up first
Data and sources to start from
Simulation for checking the formulas, plus real scores from your own matches or a published league or tournament to estimate a rally chance and test the predictions.
References to cite
- Feller, W. (1968). An introduction to probability theory and its applications (Vol. 1, 3rd ed.). Wiley.
- Ross, S. M. (2019). A first course in probability (10th ed.). Pearson.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
rallies are not all equal- Assuming the chance of winning a rally is the same for the server and the receiver. In many sports it is not.
- Forgetting the win-by-two rule, or applying it from the start instead of from the tied score.
- Using the 21-point formula for badminton without saying that the real game stops at 30.
- Applying the model to pickleball's traditional side-out scoring, where only the server can score.
- Simulating a few hundred games and calling the formula confirmed.
- A simulation that has the same mistake as the formula: check one case by hand.
More Maths IA ideas like this
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Hot streaks: is a player streaky or just luck?●●○
Rating players: how does the Elo system work?●●○
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.