Does it make sense to play the Powerball now?

Generally, lotteries are a tax on people who are bad at math.
Today, however, it seems like it might make sense to buy a ticket. The odds of winning are something like 1/300,000,000 but the jackpot is over 1 Billion dollars, so at \$2/ticket, the expected return is greater than 1.

So, should I play?

Sure. Play these numbers: 1,2,3,4,5, powerball 6. They have the same chance as any other combination, don’t they?

It makes sense to spend an insignificant amount of money on a lottery anytime. As soon as the amount of money is significant it doesn’t make sense.

You need to calculate based on the after-tax jackpot, first of all. Then you need to account for the possibility of multiple winners. And finally you should factor in the diminishing marginal utility of money. Based purely on numbers I suspect it’s still a bad wager.

Do you enjoy daydreaming? A Powerball ticket is just daydream fuel. For a few days, you have a shot at untold riches, and when you inevitably lose it’s not even disappointing because the odds were so stacked against you.

The diminishing marginal value of money is the big one, here. Yes, on the one hand, 900 million dollars is a heck of a lot of money. It’s ten times as much as, say, 90 million dollars. And so the expected value, in purely monetary terms, of a ticket for a 900 million dollar jackpot is ten times as much as the expected value of a 90 million dollar jackpot.

But now think about what 900 million dollars actually means. That’s enough to live out the rest of your life in luxury, without ever needing to work again, and while splurging on things you really like… but then, \$90 million is enough for that, too. So really, the prize is hardly any bigger now than it is when it’s \$90 million. And thus, if a ticket wasn’t worthwhile then, it’s not worthwhile now, either.

On the other hand, you could join a pool with nine other people, and buy ten times as many tickets, and agree to split the winnings if any of the tickets win. In this case, the prize amount is exactly the same as it was at \$90 million, but now the odds are ten times as good. And maybe that is worth it. The pool doesn’t change the monetary expected value, but it does change the worth.

Except most likely you’ll be splitting the pot with a bunch of other winners. I don’t know about Powerball specifically, but in other lotteries, that’s been one of the most popular combinations. For example in the UK lottery, something like 10,000 people play it.

If you are asking whether the expected value of a ticket exceeds its purchase price, it’s hard to say. To value the ticket, start with the value of the cash prize, not the annuity, so you know the real present value of the prize. It’s easy to say the odds are roughly 1 in 300 million and the ticket price is \$2 so if the present value of the prize is \$600 million, the expected value is \$2. But that estimated expected value is wrong because it fails to account for the additional possibility of winning a lesser prize. I think those lower prizes add about 20 cents to the expected value. More important though, it fails to account for the possibility that you will buy a ticket whose numbers match somebody else’s numbers and you have to share the prize. The likelihood that your ticket matches someone else’s is a hard-to-estimate function of the number of tickets sold and the randomness of the distribution of the picks (others and yours).

The tv news said 80% of jackpot winning tickets were numbers randomly selected by computer. I will infer that means that the inverse, 20%, of ticket buyers select their own numbers in drawings (there is a logical weakness in this estimate I’m ignoring but I won’t dwell on it because I decided don’t want to research the actual number of manual vs. automatic number selections).

Those 20% of tickets have a lot of commonality. People don’t pick numbers randomly. They usually pick specific numbers for some reason. One lottery result I remember reading from Massachusetts involved picking six numbers on a six-by-six grid (1-36). The ticket from had six grids you could choose numbers on. One week, the numbers drawn were, I believe, 6, 11, 16, 21, 26, 31. There were like 20 winners and the grand prize was worth very little when shared. Why so many winners? Those winners selected numbers by picking six tickets off the grid- the top row, the bottom row, the left column, the right column and the two diagonal lines. One of the diagonals won. Many more people play numbers based on significant dates. If the next Powerball numbers are a variation of the dates that the Red Sox won the world series, I’ll bet the winners would be clustered around Boston. Even personally significant dates are likely to cluster. Lots of numbers 1-12, more clustering of 1-30, etc.

So to reduce the chance of sharing your prize and maximize the value of your ticket, you’d have to consciously avoid numbers that others might have picked manually. How much that adds to the value of a ticket is hard to guess. There might be additional value if you design a better random number generator than the lottery’s picking system so you can avoid any clustering their algorithm introduces, but i don’t know anything about potential weakness in their randomizer.

Ironically, taxes don’t really reduce the expected value of your prize. You can deduct your gambling losses from your winnings. If you guaranteed your win by buying every number combination, you would pay taxes only on the amount of the prize in excess of your cost to purchase the tickets. But systematically buying 300 million tickets and then sorting through them to find all the lower valued winners too would be a logistical nightmare. Furthermore you’d still face the risk of sharing the prize which could wipe you out.

Not quite. Your expected value ignores the risk of having to share the prize, which risk increases with bigger prizes because more tickets get sold. It also ignores the value of lesser prizes.

This. I can’t retire on a million dollar win so the fantasy doesn’t work as well. \$600 million dollar after tax win is a true fantasy that \$2 fuels nicely for a couple of days. It would be fun to retire comfortably and play Santa Clause the rest of your life.

Somebody will win. Only way to win is play.

Most of the time, that’s a way, on average, to lose money. Generally, the expected return is less than the ticket cost.

This time, that may not be the case, but I’m still not going to play, since even if I won, I suspect it would ruin my life.

Exactly. A 2 buck bet is an investment in cheap entertainment. Days before the lottery are spent counting the winnings. Then if you do lose, you actually win.

Surveys of previous big winners indicate that their lives were happier before the big win.

Of course, maybe you could beat those odds, too?

Gambling does nothing for me, but when the jackpots get huge I’ll spend a buck or two just for the heck of it. The odds of winning first prize are always about 1:300,000,000 (or about a 0.000000003% chance) regardless of the jackpot size or the number of people playing (these have absolutely no effect on the odds). But if you don’t play at all you have an absolutely zero percent chance of winning…

What’s the expected return on an \$8 movie ticket?

Unless the jackpot rolls over…

I disagree. The chance of any one number coming up is the same as another, but the example you stated is not “any number.” A specific number with a human designated meaning has a lower probability than one without a human designated meaning.

For instance, the chance of any combination of five number is the same as any other combination, but you have added a specific condition. We as humans say the number two comes after the number one.

Then you say the number three comes after the number two. That is still another condition.

So right there you’ve added two more conditions thus the odds go up a bit higher, than if they were just any five numbers chosen willy nilly for no reason.

So the numbers 1,2,3,4 and 5 would have more conditions thus lower odds.

I’ve been thinking about this, too. Whoever gets the money stands an excellent chance of having their life profoundly messed up, even if they’re reasonably well-grounded, and good with money.