Maths IA · Calculus · topic 25 of 97
How much should you bet or invest? The Kelly criterion · IB Maths IA idea: guide, sample graph and step-by-step plan
the answer is a surprise, then test it!You have a bet where you win more often than you lose. How much of your money should you put on it? Too little wastes the edge, and too much can ruin you even though the odds are in your favour.
- ✓ you like optimisation problems with a surprising answer
- ✓ you want to use logarithms and differentiation on something other than a box or a can
- ✓ you enjoy testing a result by simulating thousands of plays
The big idea
Repeating a bet turns wealth into a product of factors, so the long-run growth rate is the average of the logarithms of those factors. For a fraction f of wealth staked, that average is a function g(f), and its derivative gives the best fraction in a line of algebra. The surprise is the shape of g: it rises, peaks, then falls through zero, so betting double the best amount can lose money in the long run. The calculus is short. The marks come from testing it: simulate thousands of plays, compare the typical path with the average path, and say what the model ignores, such as odds you can only estimate and money you cannot split exactly.
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 →
Write the expected log growth g(f), differentiate and solve g'(f) = 0.
→ A logarithmic function and a rational equation. Prove the answer is a maximum, and check it against a table of values.Solve g(f) = 0 for the stake above which wealth shrinks in the long run.
→ An equation with no neat solution, so you use technology. Compare the root with twice the best fraction.Simulate thousands of runs of several hundred bets at half, full and double the best fraction.
→ A random model checked against theory. Compare the median path with the mean path and explain why they differ.Change the odds so a win pays b times the stake, and see how the best fraction depends on b and p.
→ A formula with two parameters. Plot it and find where it says to bet nothing.Suppose your estimate of p is wrong by a few percent. Find what that does to growth.
→ Sensitivity analysis. Compare an over-estimate with an under-estimate of the same size, check which costs more growth, and explain why.Replace the win-or-lose bet with a share that rises or falls by fixed percentages, or with real return data.
→ A new g(f) built from your own assumptions. Say how realistic the model is.Your Maths IA research question: run the check
which bet? which numbers?Write your own question first. Then make sure it ticks every box:
- Names the bet or investment: its probabilities and payoffs, as assumed numbers with a reason.
- Says what is optimised: the long-run growth rate of wealth, and why that and not the average gain.
- States what the simulation tests: the number of plays, the number of runs and what you compare with the theory.
- Goes beyond the formula: there is a derivation to make and a surprise to test, such as over-betting.
- Says what you leave out: costs, taxes, and the fact that the odds are not known exactly in real life.
The model
Stake a fraction f of your wealth. A win adds b times the stake, a loss removes the stake. Draw the two outcomes for your own bet.
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.
What the bet or investment is, what a good answer would be, and your prediction for the best fraction.
Wealth as a product of win and loss factors, the number of wins as a binomial, and why growth uses logarithms.
g(f), its graph, and a table of values checked by hand.
The derivative, the solution, a proof of the maximum, and how the answer changes with p and b.
The break-even fraction, the loss of growth from betting double, and the same facts seen in simulated runs.
How an error in your estimate of p changes the growth, and a simple rule you recommend.
Back to your aim: the best fraction, how fragile it is, and what a sensible bettor would do.
Known odds, independent bets, no costs, money that divides perfectly. Say which assumption matters most.
Where every formula and fact came from.
What your Maths IA graph could look like
mark the peak AND the zeroA made-up bet to show the form of a strong graph, not the answer: win chance 60%, even odds, growth in percent per bet. Yours will look different. example only
- The curve rises, peaks and then falls below zero. That shape is the whole story.
- Axis labels give units: fraction of wealth, growth in percent per bet.
- Calculate the peak and the zero crossing, then mark both. Reading them off the graph is not enough.
- Plot growth rates measured from your simulation as points: close agreement with the curve is your evidence.
- Another graph to consider: simulated wealth on a log scale for half, full and double the best fraction.
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 what a favourable bet is. Before any algebra, guess whether staking everything each time is wise, and what fraction you would pick.
- Fix the betChoose the win chance and the payoff, with a reason. Say what a win and a loss do to your wealth.
- Wealth after n betsWrite the wealth as a product, use the binomial distribution for the number of wins, and see why logarithms turn the product into a sum.
- Trial run (1 lesson)Build a 20-bet spreadsheet simulation for a few fractions, and compare the average of the log growth with g(f).
- Find the best fractionDifferentiate g, solve g'(f) = 0 and show it is a maximum. Check against a table.
- Find the break-even fractionSolve g(f) = 0 with technology and compare the answer with twice the best fraction.
- Simulate properlyChoose the number of bets and runs, fix a random seed, and compare typical paths with the average path.
- Go further, with a reasonTest an error in p, or uneven odds. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means in the real setting and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Binomial distribution – Model the number of wins in bets as and write wealth as .
- Logarithms and expected value – Take logs so wealth becomes a sum, and write the growth rate as the expected log factor.
- Differentiation and optimisation – Solve to get and use to show a maximum.
- Solving equations with technology – Find the break-even fraction where numerically.
- Simulation – Simulate many runs, plot wealth on a log scale and compare medians with the theoretical growth.
Key equations
Syllabus links
- 1.7Rational exponents and laws of logarithms
- 2.9Exponential and logarithmic functions
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 5.6Standard derivatives; chain, product and quotient rules
- 5.8Maxima, minima, inflexion and optimisation
- 1.5Exponents and an introduction to logarithms
- 1.9Laws of logarithmsHL
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 5.6Stationary points
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
Read up first
Data and sources to start from
Simulate repeated bets with a random number generator and a fixed seed; optionally compare with a real sequence of returns or odds.
References to cite
- Kelly, J. L., Jr. (1956). A new interpretation of information rate. The Bell System Technical Journal, 35(4), 917-926.
- MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (Eds.). (2011). The Kelly capital growth investment criterion: Theory and practice. World Scientific.
- Poundstone, W. (2005). Fortune's formula: The untold story of the scientific betting system that beat the casinos and Wall Street. Hill and Wang.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
mean wealth ≠ typical wealth- Maximising the average wealth instead of the average of its logarithm. The first says bet everything, which almost surely ends in ruin.
- Using a percentage where the formula needs a decimal, or the reverse.
- Presenting one simulated path as evidence. A single run can look good or bad by luck; you need many.
- Trusting the best fraction when p is only an estimate. Check whether over-estimating or under-estimating costs you more.
- Forgetting that the formula assumes you can bet any fraction and that the odds never change.
- Reporting the mean of simulated final wealth and ignoring the median, which is what a typical bettor sees.
More Maths IA ideas like this
Two shares or one? Diversification and portfolio risk●●○
Lotteries: is a ticket ever worth buying?●●○
When should you stop looking? The 37% rule●●●
Pricing an option: from coin flips to Black-Scholes●●●
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.