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Why do people buy lottery tickets with negative expected value?

Start with the ledger for one KRW 1,000 Lotto 6/45 game. Donghaeng Lottery presents the split of sales as 50% for prizes, 43.6% for the Lottery Fund, 5.4% for retailer commission and 1.0% for other operating costs. Prizes are half of sales, so across all buyers combined, every KRW 1,000 paid returns an average of KRW 500 in prizes. The other KRW 500 does not come back. This page looks inside that calculation and then summarises what economics and psychology offer as reasons people buy anyway. It neither condemns nor recommends buying.

Where the KRW 500 comes from

The KRW 500 average is not only a promise; it shows up in results. Multiply winners by prize amounts for each prize in the 1,160 rounds from 88 to 1247 and add them up, and prizes are 50.0% of sales. By year, only 2005 differs (50.05%); every other year is 50.0%. The ratio is a stable structure. The KRW 500 returned per KRW 1,000 game breaks down by prize as follows.

Average return of a KRW 1,000 game (rounds 88–1247, before tax)
PrizeChance one game wins itContribution to the average return
1st1/8,145,060about KRW 238.7
2nd6/8,145,060about KRW 39.8
3rd228/8,145,060about KRW 39.8
4th (fixed KRW 50,000)11,115/8,145,060about KRW 69.8
5th (fixed KRW 5,000)182,780/8,145,060about KRW 112.0
Total2.38% chance of any prizeabout KRW 500

Notice that the first prize contributes about KRW 239, nearly half of the KRW 500 average, yet your chance of it is one in eight million. For most people who buy a game, the KRW 500 average is therefore an average drawn from something that does not happen to them. A game wins any prize with probability only 2.38%, most of that being the KRW 5,000 fifth prize (2.24%). 97.6% of games return nothing.

Tax matters too. Fourth and fifth prizes are under the tax minimum, but prizes above KRW 2,000,000 are withheld at 22% or more. The figures on this page are before tax; tax is covered in the South Korea lottery prize tax guide.

What a year of buying looks like

Buying five games (KRW 5,000) a week for a year costs KRW 260,000. On average KRW 130,000 comes back as prizes and KRW 130,000 does not. That KRW 130,000 is an average: some people recover tens of thousands of won through a few fifth prizes in a year, others receive nothing. The average describes the pooled result of many buyers, not what will happen to you.

One economic answer: expected value alone does not explain choice

Is a choice with negative expected value irrational? Economics does not necessarily say so. Friedman and Savage (1948) tried to explain people who buy both insurance and lottery tickets through the shape of the utility function. As we understand it, the idea is that the curvature of utility can vary with wealth, so the same person could choose insurance (risk aversion) and lotteries (risk seeking). We confirmed only the paper's citation and secondary descriptions, so we do not present its model in detail. The point worth keeping is that this account avoids the simple reading "buying lottery tickets is an arithmetic mistake".

Another explanation is enjoyment relative to cost. A lottery ticket can be seen as a product that buys a few days of imagining "what would I do if I won". If the value of that imagining exceeds the expected loss, the purchase may be a reasonable entertainment expense, like a cinema ticket or a game fee. On this view what matters is whether the spending stays within an entertainment budget.

A psychological answer: small probabilities feel larger

Kahneman and Tversky's prospect theory (1979) holds that people do not use probabilities as given but convert them into "decision weights". According to the abstract, decision weights are generally lower than the corresponding probabilities, except in the range of very low probabilities, where they exceed them. The authors say this overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling. One in eight million is practically zero as arithmetic, but in the mind it is treated as "something other than zero". We confirmed only the abstract as reproduced on a secondary page.

This overlaps with "ignoring size" in Probability illusions. If you cannot tell small probabilities apart, 1 in 8 million and 1 in 1 million feel alike, and winning feels closer than it is. The availability heuristic adds to it: jackpot news is visible; tens of millions of losing tickets are not.

Other reasons

What to take from this

The conclusion is neither "don't buy" nor "it's fine". It is to keep two facts clear.

Limits: the 50% is the average over all buyers, not a figure for a particular person or round. When the jackpot rolls over, the expected value of a particular round can differ; that question belongs to the rollover expected-value calculator planned for this site. The economic and psychological accounts are hypotheses and do not prove why any one person bought. We checked only citations and abstracts of the papers.

What actually happens when you buy five games

An average is a sum over many people. Suppose you buy five games (KRW 5,000) once; here is the distribution of what actually happens. A game wins some prize with probability 2.3834%, most of that being the KRW 5,000 fifth prize.

Outcome of one round with five games (KRW 5,000), before tax
OutcomeProbability
All five games lose (KRW 5,000 lost)about 88.6%
At least one game winsabout 11.4%
Only one fifth prize (KRW 5,000 back, break-even)most of those wins
To recover the cost or moreat least one fifth prize (probability about 11.4%)

Buying five games once means about an 88.6% chance of losing the whole KRW 5,000. The average expected return is KRW 2,500, but the actual result is nearly always zero, and occasionally KRW 5,000 or more. The table makes concrete the point that the KRW 2,500 average is not "what happens to me" but "a figure made by the sum of many people". Even someone who buys five games every week for 52 weeks has about a 0.2% chance of never winning in any week of the year, so a year with no win at all is rare. It matters, though, that most such wins are fifth prizes of KRW 5,000.

Questions that come up

Does a rollover make the expected value positive? This page deals with the structure of an average round. A rollover can raise the expected value of its round by enlarging the first prize, but the risk of sharing with more winners and the tax grow too. How to calculate a rollover round belongs to a separate calculator. Here, please note only that the 50% is the prize-to-sales ratio.

Is buying a lottery ticket gambling? This page does not argue over terms. It deals only with the fact that this is a game of chance returning half of sales on average. Responsible-play information is on its own page.

How much is reasonable to spend? A website cannot decide that. It is generally advised to set a limit beforehand at an amount whose loss would not affect daily life and to stay within it.

LottoFlick is independent and not an official lottery operator. Numbers are random and cannot be predicted.

Sources and verification