A clever argument for buying lottery tickets

That seems like a possible candidate for the problem: Perhaps my round figure 5.5 years was a bit too round. Let us do the calculations. For simplicity I'm going to assume that interest works like this: On December 31st of each year, I put my earnings for the year into a savings account, which then grows by a factor 1.04. This is not realistic, of course, but it does seem that the answer ought not to depend on details of the model of interest. So, consider how much I have to work in the case where I buy the ticket and don't win. At the end of the first year, I put $159900 in my account, which grows to $166296 by the magic of 4% interest. Then it grows thus:

End of 2nd year: 339347.84 End of 3rd year: 519321.75 End of 4th year: 706494.62 End of 5th year: 901154.41

and I need to work another 988.46 hours to reach the million. Now consider the case where I didn't buy the ticket. Then I have 901276.07 at the end of the fifth year, and must work another 987.24 hours. The difference is 1.22, which as you suggested is larger than the 1.1.

A cross-check: If I buy the ticket, then there's a 99.99% chance that I have to work those (5 years + 988.46 hours), which is 10987.36 expected hours. If I don't, there's a 100% chance that I have to work (5 years + 987.24) hours, or 10987.24 expected hours. So the ticket is costing me 0.12 hours, the same as we found above.

I repeated the calculation on the assumption that my expenses are $35k and $45k a year, and both are even worse for the lottery-buyer, which surprises me a bit. I don't see an obvious way to change the numbers, then; although I could still fall back on the argument that leisure hours at age 30 are worth more than at age 35. It presumably does not take a whole lot of weighting to overcome the 0.12-hour disadvantage of the lottery. But my initial numbers were clearly flawed due to not calculating exactly enough what was needed at the end of the period. Thanks for clearing up my confusion.

OK, so let's look at actual numbers.

I earn around £28,000 per year. Currently, I can't save any money - in fact I have to find extra sources of income to meet expenses for me and my wife.

Accounting for holiday time, and so on, I earn £15 an hour, give or take, so a £1 lottery ticket is worth four minutes of my work time.

Currently, I do not have any savings, or any prospect of saving any money. Thus, I expect to work for at least the next thirty-five years, or 64,400 hours.

If I won the average £2,053,984 prize for the UK National Lottery jackpot, I would not only be able to retire, thus saving those 64,400 hours, if I stuck that in a bank at 5% interest I would get £102699.20 a year - nearly four times my current salary. So I could actually be saving 239,631 'hours' over that 35-year period.

The odds of winning the national lottery jackpot are 1 in 13,983,816, which would mean my expected winning (only counting the jackpot, not the lower prizes) would be 0.017 hours, or 1.02 minutes.

That suggests that for me, at a relatively high wage, the lottery is still irrational to play. However, it also suggests that for anyone whose earnings are below £3.82 per hour (for example someone working a minimum wage job but also doing unpaid overtime or an intern) it could be rational to play the lottery.

The cost of a lottery ticket is a lot more than the jackpot times the likelihood of winning.

In lotteries with rollover jackpots, lottery tickets can become positive expected value... as long as you don't take into account the possibility of having to split the jackpot with another winner.

I picked numbers that are easy to work with to illustrate a theoretical point. Whether any actual lottery can do the same thing for the particular situation of any real person is a separate issue. Again, the part of the lottery ticket's price that covers overhead and profit is only relevant in the context of real lotteries. The question I'm asking is whether it is possible for a bad or neutral deal in dollars to become a good deal in hours of life.

I am reasonably convinced that I did not do so; when I said "roughly 5.5 years to accumulate a million", I was taking compound interest into account.

somewhere you should be missing something

Fine, but where and what? I've already gone through the math a couple of times looking for errors; the purpose of posting here was to get fresh eyes on the problem.

One could also argue that the state and its schools would be able to do a much better job if they were better funded, arriving at the opposite conclusion.