Maths IA · Statistics and probability · topic 46 of 97
Penalty kicks: where should you shoot? · IB Maths IA idea: guide, sample graph and step-by-step plan
a table, two equations, a decisionA penalty is over in less than a second, so the keeper has to move before seeing the ball. If the kicker always shoots to one side, the keeper learns it. So how often should each of them go left?
- ✓ you watch football and wonder why keepers dive before the kick
- ✓ you like probability that has a decision at the end of it
- ✓ you can collect penalty data for a team or a league
The big idea
Reduce a penalty to a table. The kicker chooses a side, the keeper chooses a side, and each pair of choices has a chance that the kick is scored. Neither knows the other's choice, so a fixed choice can be exploited. The way out is to choose at random with the right probabilities. Find them by making the other player indifferent: the mix at which the keeper's chance of saving is the same whichever way they dive. That is a pair of linear equations and an expected value. The method is short, so the marks come from the data: where your numbers come from, how well real kicks match the prediction, and what the table leaves out.
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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Write the kicker's expected scoring chance for each choice as a function of the keeper's mix, and find where the two are equal.
→ A pair of linear equations in the two mixing probabilities. You must say why equal scoring chances is the right condition.Suppose the keeper dives one way more often than the equilibrium says. What should the kicker do, and how much do they gain?
→ A graph of two straight lines with a crossing point. It shows why any departure from the mix can be punished.Count where kickers aimed and where keepers dived in a set of real penalties, and compare the proportions with the equilibrium.
→ A real comparison of data with a model, which can use a chi-squared test (AI) or a confidence interval for a proportion.Add the centre as a choice for both players.
→ A 3x3 table. You may need to solve a larger system or use technology, and you say whether the centre ever pays.Give left-footed and right-footed kickers different tables, or a strong-side and a weak-side kicker.
→ The equilibrium moves with the table. You compare two players and say what drives the difference.Find the scoring chance both players settle on and compare it with the chance if the kicker always chose their better side.
→ A number that shows what randomising is worth. You discuss how large the gain is against the uncertainty in the table.Your Maths IA research question: run the check
whose kicks? how many?Write your own question first. Then make sure it ticks every box:
- Names the table: which choices count, and where the scoring chances come from (data, not guesses).
- Says what you calculate: the equilibrium mix, the value of the game, or how far real play is from it.
- Needs an equation to solve and a decision to justify, not only a count.
- Says how real penalties will be compared with the prediction, and what would count as a difference.
- States what you leave out: the centre, kick height, pressure, the keeper's guess from the run-up.
The model
Two players, two choices each, and a table of scoring chances. Draw yours with your own sides and your own numbers.
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.
The competition or team you chose, why penalties suit a game model, and what you expect before calculating.
How many kicks, where they came from, how you decided what counts as a side, and what you left out.
Scoring chances for each pair of choices, with how reliable each one is, given how many kicks it comes from.
The two indifference equations, solved, and the value of the game. Every step defined.
A graph of the kicker's scoring chance against the keeper's mix, with the crossing point marked.
Predicted and observed proportions side by side, and a test of whether they differ by more than chance.
Back to the aim: how often to shoot each way, and whether real players come close.
Small samples, the centre, kick quality, the keeper's choice not being independent of the kicker's. Which matters most?
Where every kick and every statistic came from.
What your Maths IA graph could look like
mark the crossing!A made-up table to show the form of a strong graph, not the answer. Each line is the chance the kick is scored if the kicker aims one way; the lines cross at the keeper's equilibrium mix. example only
- Two straight lines on one graph: the crossing point is the equilibrium, and you can see it.
- Axis labels say what q means, and the y-axis says what is being measured.
- Calculate the crossing point first, then mark it. Reading it off the graph is not enough.
- Each estimate comes from a small number of kicks. Show that with error bars on your points.
- Another graph to consider: observed shares of kicks to each side next to the equilibrium shares.
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 out how penalties work and why the keeper must move early. Before any calculation, guess how often a kicker should go to their natural side.
- Choose the dataPick a team, league or tournament with enough kicks, record where each was aimed and where the keeper dived, and say how you coded them.
- Build the tableWork out the chance of scoring for each pair of choices, and how many kicks each cell is based on.
- Trial run (1 lesson)Do the table for 20 kicks first. Are the categories clear? Are some cells empty? Fix the coding now.
- Find the mixWrite the expected scoring chance for each choice and solve for the mix that makes the other player indifferent. Check by substitution.
- Draw the best repliesGraph the two lines against the keeper's mix and mark the crossing point and the value of the game.
- Compare with real playPut observed shares of kicks beside the equilibrium shares, and test whether the gap is larger than chance would give.
- Go further, with a reasonAdd the centre, or compare two kickers. 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 kicker and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every symbol is defined, the data source is named, and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Conditional and joint probability – Compute the chance of scoring for each pair of choices from counts, and the chance of each pair under independent mixing.
- Expected value – Write the kicker's expected scoring chance for each choice as a linear function of the keeper's mix.
- Linear equations – Set the two expected values equal and solve for the mixing probabilities, then check by substitution.
- Graphs of straight lines – Draw the two best-reply lines, find their intersection and read the value of the game from it.
- Goodness-of-fit testing – Compare observed counts of kicks to each side with the counts the equilibrium predicts; check expected counts and report the p-value.
Key equations
Syllabus links
- 2.1Straight lines
- 4.5Basic probability and expected frequency
- 4.7Discrete random variables and expected value
- 4.11Conditional probability and independence
- 4.5Basic probability and expected frequency
- 4.7Discrete random variables and expected value
- 4.11Hypothesis tests: chi-squared and t-test
- 4.6Venn and tree diagrams, conditional probability
Read up first
Data and sources to start from
Record where each penalty was aimed and where the keeper dived from match video or a league's published shot data; your own team's kicks work too if there are enough.
References to cite
- Palacios-Huerta, I. (2003). Professionals play minimax. The Review of Economic Studies, 70(2), 395-415.
- von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton University Press.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
small samples fool you- Calling the equilibrium the best thing to do against any keeper. It is only safe against a keeper who plays well.
- Equating the two probabilities p and q, or mixing up whose indifference gives which equation.
- Building the table from a handful of kicks and treating the chances as exact.
- Using a quoted table from a paper without saying whose data it is and how many kicks it covers.
- Ignoring that the keeper's dive and the kicker's aim are not always simultaneous: some keepers wait.
- Testing for a match with a chi-squared test without checking the expected counts are large enough.
More Maths IA ideas like this
Tie-breaks: do league rules change who wins?●●○
Goals and tries: are they random?●●○
From rally to match: how small edges become big wins●●○
Hot streaks: is a player streaky or just luck?●●○
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.