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Maths IA · Statistics and probability · topic 44 of 97

Lotteries: is a ticket ever worth buying? · IB Maths IA idea: guide, sample graph and step-by-step plan

expected value is an average, not a promise

A 6-from-49 lottery has 13,983,816 possible tickets, so one ticket has a chance of about 1 in 14 million. Is there a jackpot big enough that the average ticket stops losing money?

Difficulty●●○ Medium
KitA spreadsheet, the published prize rules of one lottery, your GDC
TimeAbout 12 to 15 hours
FitsAA SL3 of 3AA HL3 of 3AI SL3 of 3AI HL3 of 3
  • ✓ you like counting problems with huge answers
  • ✓ you want to know what "expected value" really tells you
  • ✓ you would rather test a claim with numbers than argue about it

The big idea

Every ticket is a random prize with a price attached, so its expected value is a sum over the prize tiers: probability times prize. Probabilities come from counting combinations. The twist is that the biggest prize is shared. If other players pick the winning numbers too, your slice shrinks, and the number of other winners is itself random. That lets you find the break-even jackpot, the size at which a ticket's expected return equals its price, and see how it moves with the number of tickets sold. You also meet risk, because two games with the same expected value can feel completely different. This is a maths investigation of an expected value, not a guide to playing: your conclusion should say what the numbers show about the odds and the typical loss, and what they leave 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.

Swipe for more twists →

Count every tier

Use combinations to find the probability of matching 3, 4, 5 or 6 of the numbers, and check they sum to 1 with the no-prize outcome.

→ A hypergeometric distribution built by hand. Show it for one real game with its actual prize tiers.
Expected value per ticket

Combine probabilities with the published prizes and compare the total with the ticket price.

→ The lower tiers return a surprisingly large share. Find which tiers matter most and which barely count.
Share the jackpot

Model the number of other winners as a Poisson (or binomial) variable and work out the expected share of the jackpot.

→ A new expected value that depends on tickets sold. Derive the average share and compare it with simply ignoring sharing.
Find the break-even jackpot

Solve for the jackpot at which expected return equals the ticket price, for different numbers of tickets sold.

→ An equation in the jackpot with a parameter. Explain why the answer is so much larger once sharing is included.
Measure the risk

Find the variance and standard deviation of the return on one ticket, and the chance of winning nothing in n draws.

→ Two games with the same expected value can have very different spreads. Say what that means for a real player.
Compare two games

Repeat the analysis for a second lottery with a different pool size or prize structure.

→ A comparison with a clear criterion. Decide which game has the better expected return and say how robust the answer is.

Your Maths IA research question: run the check

which game? which prizes?

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

  • Names the game: how many numbers are drawn from how many, and its published prize tiers and ticket price.
  • Says which quantity you will calculate: the expected return per ticket, the break-even jackpot, or the spread of outcomes.
  • Says how sharing is handled and where the number of tickets sold comes from.
  • Goes beyond a single multiplication: there is a model to build and a result to test.
  • States what you leave out: tax, rollovers, how jackpot size changes sales.
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The model

A fixed set of prize tiers, plus one jackpot that has to be shared with a random number of other winners. Redraw it for your own game, with its real odds.

Choose 6 numbers from 49. Six are drawn.Bar length = number of digits in the odds (log scale)match 31 in 57match 41 in 1,032match 51 in 54,201match 61 in 13,983,816Jackpot J, shared by you and K other winners:each gets J / (1 + K)K ~ Poisson(λ), λ = S·pS = tickets sold, p = 1 / 13,983,816the jackpot is the onlytier that is sharedmore tickets sold, smaller share

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
Which game, who plays it, and what you expect before calculating: is a ticket worth its price?
Counting the tiers~2 pages
Combinations explained and used for each prize tier. The probabilities add up, with a check.
Expected value of a ticket~1.5 pages
Every prize and probability in one table, the sum, and a comparison with the price.
Sharing the jackpot~2 pages
The model for other winners, the average share derived, and the effect of tickets sold.
Break-even jackpot~1.5 pages
The equation, its solution for several sales levels, and a graph of expected return against jackpot.
Risk and variance~1.5 pages
Variance of one ticket, the chance of winning nothing over many draws, what the spread means.
Conclusion~1 page
Back to your aim: what the expected value and the break-even jackpot show, with numbers.
Evaluation~2 pages, the biggest part!
Fixed prizes against pooled prizes, tax, rollovers, sales that depend on jackpot size. Say which assumption matters most.
Bibliographynot counted
Where every prize rule, price and sales figure came from.

What your Maths IA graph could look like

mark the break-even jackpot!

Made-up numbers to show the form of a strong graph, not the answer: a game with a price of 2 and fixed lower prizes. Yours will use your own game's rules. example only

0102030405060708001234Jackpot (millions of currency units)Expected return per ticketjackpot never sharedshared with other winnersticket priceyour own computed values (made-up)break-even jackpot:where the curve meets 2ILLUSTRATIVE · MADE-UP DATA
  • Both versions on one set of axes, so the effect of sharing is visible at a glance.
  • Axis labels give units: jackpot in currency units, return per ticket in the same units.
  • Calculate the break-even point exactly, then mark it. Reading it off the graph is not enough.
  • Plot values from your own table as points: agreement with the curve is your evidence the model is right.
  • Another graph to consider: the chance of at least one jackpot against the number of draws played.

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.

Left: straight lines for expected return against jackpot, steeper when the jackpot is never shared, with break-even points where each meets the ticket price of 2. Right: an S-shaped curve on a log axis showing the chance of winning at least one jackpot against the number of draws played.
(a) Expected return per ticket against the jackpot, price 2: break-even is 23 million with no sharing, 37 million with one other winner on average. (b) The chance of a jackpot reaches 50% only after 9.7 million draws.Source: Computed from the model; the lower prizes (10, 100 and 3,000) are assumed, not a real game's.

Your game plan

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

0 of 10 done

  1. Read and predictFind the rules of one real lottery. Before any calculation, guess whether its expected return is above or below the ticket price, and by how much.
  2. Choose the gamePick one game with published odds and prizes and say why: the one near you, or the one with the biggest jackpots.
  3. Count the outcomesUse combinations to find the number of tickets and the number that win each tier. Check the probabilities sum to 1.
  4. Trial run (1 lesson)Calculate the expected value ignoring sharing, then check it with a simulation of a million tickets. Fix any mismatch now.
  5. Model the sharingChoose a model for the number of other winners, derive the expected share, and say what sales figure you assume and why.
  6. Solve for break-evenSet expected return equal to the price and solve for the jackpot. Repeat for three or four sales levels and describe the pattern.
  7. Measure the riskFind the variance of one ticket's return, and the chance of winning nothing across many draws. Say what the spread means in plain words.
  8. Go further, with a reasonCompare a second game, or let sales rise with the jackpot. HL: this is where the rigour for criterion E comes from.
  9. Reflect as you goAfter every result, write one line on what it tells a player and what it assumes. These lines become your evaluation.
  10. 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

  • Combinations – Count tickets and the tickets matching exactly numbers: .
  • Expected value – Sum probability times prize over the tiers, subtract the price, and compare the result with the price for different jackpots.
  • Binomial and Poisson models – Model the number of other winners as or Poisson and sum the series for .
  • Solving for a parameter – Solve for the break-even jackpot , with the share factor depending on tickets sold.
  • Variance and risk – Compute for one ticket, and the chance of no win in draws as .

Key equations

Syllabus links

  • 1.9Binomial theorem
  • 4.5Basic probability and expected frequency
  • 4.7Discrete random variables and expected value
  • 4.8Binomial distribution
  • 4.14Variance; continuous random variables and pdfsHL

Read up first

Data and sources to start from

Use one lottery operator's published odds, prize tiers, ticket price and recent draw sales figures; run a simulation to check your formula.

References to cite

  1. Ross, S. M. (2019). A first course in probability (10th ed.). Pearson.
  2. Haigh, J. (2003). Taking chances: Winning with probability (2nd ed.). Oxford University Press.
  3. Singapore Pools. (2026). TOTO. https://www.singaporepools.com.sg/

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

count the tiers separately
  • Treating a very large expected value as a reason to play. Expected value is an average over many tickets, not what one person gets.
  • Using the headline jackpot as if you always keep all of it. Other winners share it.
  • Forgetting that the ticket price comes off the return, so a return of 0.60 per ticket is a loss of 1.40 on a price of 2.
  • Adding probabilities of tiers that overlap, or counting a 5-number match inside the 6-number tier.
  • Presenting simulation output as proof. It only agrees or disagrees with your formula.
  • Writing as if the investigation recommends playing. Your job is to report what the numbers show.

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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