Once a lottery like Mega Millions has a high enough jackpot so that the expected value of the ticket is higher than the cost, why does it still feel like it’s a bad investment to buy a ticket?
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Let’s simplify things and consider a lottery that just has one big jackpot and no other prizes. Let’s say the odds of winning this jackpot are 1 in 300 million (that’s roughly the [odds](https://www.megamillions.com/How-to-Play.aspx) for getting all the numbers right to win the Mega Millions jackpot). Instead of drawing a set of numbers, let’s say you just roll a 300 million-sided die and you win the jackpot if it comes up ‘1’. Let’s say the jackpot is $1 billion and it costs $2 for each roll of the die.
Your expected winnings per roll are $1 billion / 300 million = $3.33. Clearly, that’s more than it costs and so statistically you expect to make a profit. That is, if you could roll the die enough times, your winnings would equal $3.33 per roll, for a profit of $1.33 per roll. For instance, if you were able to play 1 trillion times, you’d almost certainly make a profit close to $1.33 trillion.
The problem, of course, is that you can’t afford to play 1 trillion times. You have neither the time nor the upfront cash to finance that. So it doesn’t matter how the statistics work out in the long run. The important thing is: is there a reasonable probability that you’ll make a return on your investment before either your time or your money runs out?
Clearly, you need to win the jackpot at least once in order to have a chance to make a profit. Note that you’re not guaranteed to make a profit once you win – if it takes too many rolls, you’ll still be making a loss. But if you don’t win even once, you’ll certainly make a loss.
So how often do you need to roll the die to win the jackpot at least once? Well, it’s never a sure thing that you’ll win, but let’s say we want to have at least an 80% probability. How often do we need to roll the die for that? The probability that you’ll wint the jackpot at least once in N rolls is 1-p^(N) where p is the probability of not winning, i.e. 1-(1/300 million). So we have to solve 1-p^(N) = 0.8, i.e. p^(N) = 0.2. We can solve this by taking the logarithm of 0.2 in base *p*. If none of that makes sense to you, don’t worry, just note that this is a mathematical way of answering the question: how many times do I need to roll this die so that I’ll have an 80% probability of winning the jackpot at least once?
The answer turns out to be over 482 million times. That’s an investment of $964 million, not to mention how much time it takes to roll a die that many times.
And therein lies the rub. You do not have the time nor the money to play the lottery enough times to have any reasonable hope of making a return on your investment. Not to mention, how are you even going to buy that many lottery tickets? They don’t stock that many at your local corner store, and I doubt that Mega Millions is going to allow you to purchase that many tickets from them directly.
Also, note that even with 482 million rolls, there’s still a 20% chance that you don’t win the jackpot, and then you’ve accumulated losses of almost $1 trillion. To really be sure of a profit, you need to be playing tens of billions of times.
**In short, the problem is that the expected profit comes from a teenie, tiny probability of winning a large amount of money. If you had a 50% probability of winning $6.66 on a ticket that costs $2, the expected profit would also be $1.33 per ticket, only then you’d only have to play like a dozen times in order to be pretty sure of a profit. So it’s not about the expected value of the winnings distribution, it’s about how uneven it is.**
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