Expected Value: The Number Behind Every Bet, Ticket, and Policy

Opening Hook

Every week, millions of people in the UK spend £2 on a National Lottery ticket. The prize fund receives roughly 45 percent of total ticket sales. So before a single ball drops, the average ticket buyer is handing over £2 and receiving back, in expected terms, somewhere around 90 pence.

That is not a bug in the system. It is the entire point of the system.

The lottery knows this number precisely. The insurance company selling you a policy knows this number for every product it sells. The casino has it printed inside a manual no customer ever reads. Expected value is the single most important number in any transaction involving chance, and the people on the selling side have spent a great deal of effort ensuring you never think about it. This unit teaches you to think about it.

The Concept

Expected value is the probability-weighted average of all possible outcomes. It tells you what a bet, gamble, or uncertain event is worth, on average, if you could repeat it a very large number of times.

The calculation is straightforward. For each possible outcome, multiply the value of that outcome by the probability of it occurring. Then add all those products together.

A concrete example. Suppose someone offers you a coin-flip game. Heads, you win £3. Tails, you lose £1. Should you play?

The expected value is: (0.5 × £3) + (0.5 × −£1) = £1.50 − £0.50 = £1.00.

The expected value is positive: £1.00 per flip. If you played this game a thousand times, you would expect to be ahead by roughly £1,000. This is a game worth playing.

Now change the numbers. Heads, you win £3. Tails, you lose £4. Expected value: (0.5 × £3) + (0.5 × −£4) = £1.50 − £2.00 = −£0.50. Negative expected value. Over many trials, this game drains money from you steadily, and someone with a negative expected value game running in their favour is called a casino.

The lottery worked out. A standard UK Lotto ticket costs £2. Matching all six numbers wins the jackpot; matching five plus the bonus ball wins £1 million; smaller prizes cascade down from there. The jackpot odds are approximately 1 in 45 million. If the jackpot stood at £5 million, the expected value of the jackpot tier alone is £5,000,000 ÷ 45,000,000, which is about 11 pence. Add in the smaller prizes, and the total expected return on a £2 ticket is around 90 pence to £1, depending on jackpot size. You are paying £2 for something worth roughly half that.

One crucial limitation. Expected value is a long-run concept. It describes what happens across many, many repetitions. It says nothing reliable about any single trial. If you play the coin-flip game from the first example only once, you will either win £3 or lose £1. You will not win the expected value of £1, because £1 is not even one of the possible outcomes. Expected value is the average of many outcomes, not a prediction for one.

This is why the lottery can honestly say the expected return is 45 percent while every individual ticket buyer either wins big or wins nothing. Expected value lives in aggregate. Individual results scatter around it.

The St Petersburg Paradox. In 1738, the Swiss mathematician Daniel Bernoulli published a paper in the proceedings of the St Petersburg Academy describing a game that breaks expected value. The game works like this: a coin is flipped repeatedly until it lands heads. If it lands heads on the first flip, you win £2. If it takes two flips, you win £4. If it takes three flips, £8. And so on, with the prize doubling for each additional flip required.

What is the expected value of this game? Work through the probabilities. The chance of heads on flip 1 is ½, paying £2, contributing £1 to the expected value. The chance of tails-then-heads is ¼, paying £4, contributing £1. The chance of the sequence taking exactly three flips is ⅛, paying £8, contributing £1. Each subsequent tier contributes exactly £1, and the game can go on forever. Add up infinite £1 contributions and the expected value of the game is infinite.

So how much would you pay to play it? Most people offered this game would pay a few pounds, perhaps £10 at most. Almost nobody would pay £100. Yet the expected value is infinite. Something is wrong.

Bernoulli’s resolution was that money has diminishing marginal utility: an extra £1,000 matters much more to someone with £100 than to someone with £10 million. The utility of wealth grows more slowly than wealth itself. When you account for this, the wild, low-probability prizes in the St Petersburg game stop adding much utility even though they add a great deal of expected money. The practical lesson: expected value calculated in raw money can overstate the value of high-variance bets, especially when the prizes involved exceed what you could actually use.

Variance alongside expected value. Two bets can have identical expected values and be completely different animals. Consider: Bet A pays £100 with certainty. Bet B pays £200 with 50 percent probability and nothing with 50 percent probability. Both have an expected value of £100. But Bet B has a non-trivial probability of leaving you with nothing, and Bet A does not.

Variance is the formal measure of how spread out the possible outcomes are around the expected value. It will get its own full treatment in Unit 1.6. For now, the point to hold is this: whenever someone presents you with an expected value figure, ask what the spread of outcomes looks like. Two options with the same expected value are not the same choice if one of them can wipe you out.

Why It Matters

The places where expected value is most consequential are exactly the places where most people never calculate it.

Lotteries. Every lottery in the world, whether state-run or commercial, is designed to have negative expected value for the buyer and positive expected value for the operator. The margin varies. The UK Lottery returns roughly 45 percent of sales as prizes. Many scratch cards return a smaller fraction. The people who design these games know the expected value to several decimal places. The people who buy the tickets are rarely thinking about it at all. The lottery industry is, in statistical terms, an expected value transfer mechanism that moves money from buyers to operators and good causes, consistently and at scale.

Insurance. Here the picture is more interesting, because insurance with negative expected value can still be a rational purchase. An insurance company charges you a premium that, on average, exceeds what it expects to pay out in claims. Otherwise, it would not stay in business. So the expected value of buying insurance is negative: you pay more in premiums, in expectation, than you receive in claims.

And yet buying certain insurance is entirely sensible. Why? Because the risk you are transferring is one you cannot absorb. If your house burns down and you are uninsured, you are financially ruined. If you pay home insurance premiums for forty years and never claim, you have spent money but you are not ruined. The negative expected value of the insurance is the price of certainty. You are not trying to make money on the transaction; you are buying yourself out of a catastrophic but low-probability outcome that your finances cannot withstand.

The same logic does not apply to every insurance product. Extended warranties on consumer electronics, for example, are almost pure expected-value extraction. The probability of a claim is low, the item is replaceable without ruin, and the premium is priced to be highly profitable. The insurance company knows this. The consumer, buying the warranty at the checkout, is usually not running the calculation.

Investment decisions. A common error in investment thinking is to focus on the most likely outcome rather than the probability-weighted average of all outcomes. A property developer might reason “I’ll probably make a 20 percent profit, so this is a good deal” without fully accounting for the probability of the scenarios where the project runs over budget, the market turns, and the return is negative. Expected value in investment decisions requires multiplying every scenario’s return by its probability and summing the results. Professionals do this routinely. Individuals almost never do.

Who benefits from expected value illiteracy? Anyone who is selling a negative expected value proposition to a large number of buyers. The lottery operator. The casino. The extended warranty provider. The spread betting firm. These businesses are, at their mathematical core, expected value arbitrageurs: they buy risk at prices favourable to themselves and sell it at prices unfavourable to you. Their entire operating model depends on the gap between the expected value they know and the one you are imagining.

How to Spot It

The most documented example of expected value manipulation in plain sight is the lottery jackpot. In 2016, the UK National Lottery changed the Lotto game, increasing the number pool from 49 to 59 balls. This roughly tripled the odds against winning the jackpot, from 1 in 14 million to approximately 1 in 45 million. The change was presented as a way to create more frequent rollovers, which would generate larger jackpots that attracted more media attention and more players. The expected value of a ticket did not improve. It got worse, because the jackpot had to grow much larger before it compensated for the longer odds, and most rollovers stopped well short of that threshold.

The tell was in what was advertised and what was not. The lottery ran extensive marketing around “bigger jackpots” and the excitement of rollovers. The changed odds against winning received no marketing at all. The word “expected value” appeared in none of the consumer communications. The actual return per pound spent was never mentioned. This is the standard pattern: the seductive large number (the jackpot) is foregrounded; the probability that produces the expected value calculation is buried in the small print or omitted entirely.

The tell in any uncertain proposition involving money is the asymmetry between what is advertised and what is true. When you are shown the potential upside without the weighted probability of every outcome, you are being sold expected value illiteracy. The question to ask, always, is: what is the probability-weighted average of all possible outcomes, including the ones being left unmentioned?

Your Challenge

A friend tells you about a competition. Entry costs £5. There is a 1 in 500 chance of winning £1,000 and a 1 in 50 chance of winning £20. All other entrants win nothing.

Calculate the expected value of entering this competition. Is entering this competition rational on expected value grounds? Does the expected value calculation alone tell you everything you need to know in order to decide whether to enter? What other factor from this unit is relevant?

No answer appears on this page. Work through it.

References

UK National Lottery prize structure and payout percentage: Gambling Commission, “Pence per pound breakdown of National Lottery sales, 1 April 2023 to 31 January 2024.” URL: https://www.gamblingcommission.gov.uk/statistics-and-research/publication/pence-per-pound-breakdown-of-national-lottery-sales-1-april-2023-to-31. National Lottery (United Kingdom), Wikipedia, section on prize structure and odds: https://en.wikipedia.org/wiki/National_Lottery_(United_Kingdom)

UK Lotto jackpot odds (1 in 45,057,474): Camelot Group, official game rules and odds, as described at https://lottery.merseyworld.com/Info/Prizes.html and verified in Fetix, “UK Lotto Prize Tiers and Odds — Complete Breakdown,” https://www.fetix.app/en/articles/uk-lotto-prize-tiers-odds

UK Lotto odds change in October 2015 (from 1 in 13,983,816 to 1 in 45,057,474): BBC News, “National Lottery: New Lotto to have more balls and bigger jackpots,” October 2015. Available at https://www.bbc.co.uk/news/business-34486903

St Petersburg Paradox: Bernoulli, D., “Specimen Theoriae Novae de Mensura Sortis,” Commentarii Academiae Scientiarum Imperialis Petropolitanae, 1738. English translation: Bernoulli, D., “Exposition of a New Theory on the Measurement of Risk,” Econometrica, Vol. 22, No. 1 (January 1954), pp. 23–36. Stanford Encyclopedia of Philosophy, “The St. Petersburg Paradox,” https://plato.stanford.edu/entries/paradox-stpetersburg/

Expected value and insurance: Khan Academy, “Comparing insurance with expected value,” https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib/v/expected-value-insurance. General treatment of insurance pricing mechanics: Winsurtech, “How does an insurance company calculate profits?” https://winsurtech.com/blog/how-does-an-insurance-company-calculate-profits/

Expected value and diminishing marginal utility (St Petersburg resolution): Bernoulli (1738/1954), as above. Further discussion: Seidl, C., “The St. Petersburg Paradox at 300,” Journal of Risk and Uncertainty, Vol. 46 (2013), pp. 247–264.