Maths IA · Statistics and probability · topic 45 of 97
When should you stop looking? The 37% rule · IB Maths IA idea: guide, sample graph and step-by-step plan
list small cases first!You will see 100 candidates, one at a time, and must accept or reject each on the spot. There is a rule that picks the single best one more than a third of the time, and its cut-off is about 37% of the way through.
- ✓ you like a puzzle where the answer is surprising and exact
- ✓ you want to turn a decision problem into a formula
- ✓ you enjoy moving from a small case to a general proof
The big idea
You can only compare each option with the ones you have already seen, and a rejected option is gone for good. A natural rule is to reject the first r options, then take the first one better than all of them. Count the orders in which this wins, and the chance is a sum of terms of the form 1/(k − 1). For large n, that sum becomes an integral and the chance is −x ln x, where x is the fraction rejected. Setting the derivative to zero gives x = 1/e, with a winning chance of 1/e. Start by listing every order for three or four options, so you trust the formula. The depth comes from the proof and from changing the rules: a cost for waiting, a different goal, or an unknown number of options.
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.
Swipe for more twists →
For n = 3 and n = 4, write all the orders in which candidates can arrive and count the wins for each cut-off r.
→ A table you can check by hand. Use it to confirm the general formula before trusting it for n = 100.Condition on the position of the best candidate and sum over positions to get P(r, n).
→ Conditional probability and a sum of reciprocals. Explain why the first r candidates must contain the best of the first k − 1.Let n grow with r/n = x fixed, turn the sum into an integral, and maximise the result.
→ A link between a harmonic-type sum and ln x, then differentiation to find the maximum at x = 1/e.Simulate thousands of random orders for several n and cut-offs and compare with the exact values.
→ Agreement with your formula is your evidence. Explain the spread of the simulated results for a given number of runs.Aim for one of the top two, or the lowest expected rank, instead of the single best.
→ A new objective and a new optimal cut-off. Compare the strategies and say which a real chooser should prefer.Make each rejected candidate cost something, or let the offers expire with some probability.
→ The best cut-off moves earlier. Model the trade-off and find it for a realistic cost.Your Maths IA research question: run the check
which decision? which assumptions?Write your own question first. Then make sure it ticks every box:
- Names the decision (hiring, flat hunting, any one-at-a-time choice) and says why the rules fit it.
- States the assumptions: known n, random order, no going back, and you want the single best.
- Says what you will calculate: the exact winning chance, the best cut-off, or both, and how you will check them.
- Goes beyond quoting 37%: there is a derivation, a limit or a changed rule to analyse.
- Says how the result would change if the assumptions failed, and what you leave out.
The model
Candidates arrive one at a time and you only ever compare with those already seen. The first r set a benchmark; after that you take the first to beat it. Draw your own order.
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.
Which choice problem, what the rules are, and what you expect the best cut-off to be before calculating.
All orders for three and four candidates, wins counted for each cut-off, and a pattern spotted.
P(r, n) derived by conditioning on where the best candidate is, with every symbol defined and checked against the small cases.
A table or graph of the best r and its winning chance for n up to 100 or more, and the pattern as n grows.
Sum to integral, the function −x ln x, and a proof that its maximum is at 1/e with value 1/e.
Simulated winning rates against exact values, with the number of runs and an estimate of the error.
Back to your aim: the best rule, its chance of success, and what it means for the decision you chose.
Unknown n, no recall, other goals, people who do not behave like random orders. Say which assumption matters most.
Where every theorem and source came from.
What your Maths IA graph could look like
prove the maximumMade-up simulation results to show the form of a strong graph, not the answer: winning chance against the fraction rejected, with the exact curve for 100 candidates. example only
- The exact curve and your simulated points on one set of axes, so agreement is visible at a glance.
- Axis labels give what is measured: a fraction on x, a probability on y.
- Calculate the maximum, then mark it. Reading it off the graph is not enough.
- Plot simulated winning rates as points: agreement with the curve is your evidence the formula is right.
- Another graph to consider: the best cut-off and its winning chance against n.
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 the problem is set up. Before any calculation, guess how many candidates you should look at before choosing.
- Fix the rulesState exactly what you assume: known n, random order, decisions on the spot, and a goal of the single best candidate.
- List small casesFor n = 3 and n = 4, list every order and count how often each cut-off picks the best. Look for a pattern.
- Trial run (1 lesson)Write a short simulation for n = 10 and compare it with your hand counts. Fix any mismatch now: ties, the last candidate, off-by-one errors.
- Derive the exact chanceCondition on the position of the best candidate and sum. Check the formula against your small cases and your simulation.
- Find the best cut-offWork out the best r for many values of n and describe how it grows. Does r/n settle down?
- Take the limitTurn the sum into an integral, maximise it, and compare the limit with your exact values for large n.
- Go further, with a reasonChange the goal, add a cost for waiting, or allow recall. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a real chooser and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Counting and probability – List the orders for small and count wins; condition on the position of the best candidate for the general formula.
- Conditional probability – Show the best of the first lies in the first with probability , then sum over positions .
- Sums and limits – Write as an approximation to as .
- Optimisation – Maximise by solving and confirm with .
- Simulation – Simulate random orders to test the formula and estimate the error from the number of runs.
Key equations
Syllabus links
- 4.5Basic probability and expected frequency
- 4.6Venn and tree diagrams, combined events
- 4.11Conditional probability and independence
- 5.6Standard derivatives; chain, product and quotient rules
- 5.8Maxima, minima, inflexion and optimisation
- 5.11Definite integrals and areas between curves
- 4.5Basic probability and expected frequency
- 4.6Venn and tree diagrams, conditional probability
- 5.5Integration as anti-differentiation; areas
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
- 5.10Second derivative testHL
Read up first
Data and sources to start from
Simulate random arrival orders in a spreadsheet or short program; optionally run a classroom experiment with numbered cards.
References to cite
- Ferguson, T. S. (1989). Who solved the secretary problem? Statistical Science, 4(3), 282–289.
- Lindley, D. V. (1961). Dynamic programming and decision theory. Applied Statistics, 10(1), 39–51.
- 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
37% is a cut-off, not a promise- Quoting 37% without deriving it. The marks are in the argument, not the number.
- Mixing up the two 37% figures: the fraction you reject and your chance of success are both about 1/e, but they are different quantities.
- Rounding the cut-off the wrong way for small n. Check r and r ± 1 against the exact formula.
- Assuming the candidates come in a random order without saying why that is fair for your real situation.
- Treating simulation output as the proof. It supports the formula; it does not replace it.
- Applying the result to real choices (dating, flats) without saying that real offers are not in random order and n is not known.
More Maths IA ideas like this
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Matching students to places: the Gale-Shapley algorithm●●●
How much should you bet or invest? The Kelly criterion●●○
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.