What do they have in common? This cartoon from reader Pliny the in Between’s website Evolving Perspectives:
I heard that near the end of the big Powerball fracas, the expected gain from buying a ticket exceeded its price. That’s the first time I’ve ever heard of such a thing–but I still wouldn’t invest!
Depends a bit whether we’re talking mean, mode or median! It’s a good example of a non-Gaussian distribution where the difference between those matters.
“I heard that near the end of the big Powerball fracas, the expected gain from buying a ticket exceeded its price.” Nope. http://thefederalist.com/2016/01/12/with-the-jackpot-at-1-4-billion-is-it-finally-rational-to-purchase-a-powerball-ticket/
The analysis in the Federalist post is interesting, but not quite right from the point of view of the statistical theory of gambling. A “fair game”, by definition, is one in which the expectation (the average payout ) is equal to the stake (the cost of participating). Post-game considerations (income tax, sales tax on anything you buy with your winnings, annoying requests for money, loss of privacy, etc.) may influence an overall subjective decision on rationality, but they are not part of the fairness of the game. The post is also just wrong in saying the federal government is “the house”– the federal government does not sponsor lotteries. Many state governments sponsor lotteries, and are thus more house-like, but, again, the fairness of a game depends on the expectation and stake, not post-game considerations. (Most state lotteries are wildly unfair, in the statistical sense, because the expectation is much less than the stake. From the house’s point of view, a fair game is a disaster, for it, on average, generates no profits for the house.)
What the Federalist post does do well is consider that the expectation is raised by other payouts besides the jackpot, and the problem of the split pot– there can be more than one winner, thus reducing the payout for a jack-pot winning number. In the event, the actual payout for each winner was a bit less than the amount for a fair game with a single winner. (With a roughly one-in-300 million chance of winning, a game with a single winner would require a payout of $600 million for a $2 stake. But, of course, the game is not restricted to a single winner– and thus the Federalist, quite properly, took account of the split pot problem.)
In “How Not to be Wrong” Jonathan Ellerberg takes on the statistics of lotteries. As expected if the goal is winning, they’re pretty grim. If the goal is entertainment and the excitement of participation, the trivial amount paid for a ticket might be worth it, if that’s what entertains you.
Religion provides entertainment, too, but at a much higher cost.
It seems to be particularly irrational to buy a a fist-full of tickets. If you have no ticket you certainly won’t win the big prize but if you buy a ticket then you have a chance – an infinitesimally small one but you might reason that for a trivial expenditure it is worth playing for the excitement of participation and the pleasure of dreaming about “what you would do if you won”. But buying a bunch of tickets increases your chance of winning by such a minute amount (ten times close to zero is still close to zero) that it makes no sense at all yet I would bet that most people buying tickets buy more than one.
…its just a government sanctioned rathole…
As the old line goes, a lottery is a tax on people who are bad at math.
It is hard to resist pontificating … oh, who am I kidding: there is a misplaced point (or possibly a missing zero) in the comic.
Pope’s waver* & Pascal’s wager go nicely together, just as I imagined it!
*Such as this: http://www.livescience.com/48563-pope-francis-supports-exorcisms.html , or this: https://en.wikipedia.org/wiki/Pope_Francis#Homosexuality .
A lottery ticket buys a few days of hope. For me, that’s worth a dollar – if I don’t have to go out of my way to buy the ticket (especially if it would make a good story – “I bought the ticket with a dollar I found in a rental car”). When Powerball went to $2, I switched to Megamillions, which is still a buck. Whenever the clerk asks if I want only one, I say, “yeah, the odds are about the same no matter how many you buy.”
I’d made up my mind a long time ago to never buy a lottery ticket almost for that reason: it would signal to me a sort of loss of hope, a leaning-on-fate kind of desperation, even if minor.
It would signal a sort of dissatisfaction with my life, however minor, and a look outside of myself to luck, rather than seeing myself as driving my own life.
I get that in the right mindset lottery tickets can be fun though. I worked for a company in which it was a “thing” sometimes to put into the lottery pot. We actually won a decent amount one time.
If I ever felt the urge to play I would save a trip to the convenience store and just feed dollar bills into my paper shredder.
“Over 500,00,000 Powerball tickets…”
Looks like either a zero got lost or a comma is out of place.
I buy maybe 1 lotto ticket a year when the jackpots get ridiculously huge. No hope of winning behind it, and usually I have to be reminded on the day of the drawing about it. No reason to buy more than one either.
Your chances of winning are infinitely greater if you have a ticket than if you don’t.
I don’t know whether the expected return was positive in this case, but I seem to remember reading somewhere about an Australian lottery where the pay-out structure was miscalculated and that allowed this … (Some group bought up a few hundred thousand tickets or something and got the expected pay off, too and made a handsome profit, apparently.)
In “How Not to be Wrong” (No, I don’t get paid for this) there’s a detailed description of how the Massachusetts Lottery was legally gamed for sure returns because the payout structure allow this to happen. After noticing the pattern of ticket buying and wins, the administrators changed the rules, but the winnings were obtained entirely by legal means and the winners got to keep them.
I believe the same happened to the Washington State Lottery in the 1990s, when the payoff carried over from a preceding period in which no one had a winning ticket, and exceeded the amount expected to be wagered. An Australian syndicate then bought up about half the tickets in a huge block — and they won. Changes were made afterward to prevent it happening again.
This is ill-remembered, so I’ve looked online to try to verify this, and so far cannot find the details.
In thinking about lotteries at various stages of my life – in particular, imagining the scenarios of what I would do if I won; I’ve come across many times where the answer would be that I would not change much at all. Thus I’ve transformed the existence of lotteries into “the lottery game” thought experiment. If you would change very little – for example, keep the same career, live in the same neighbourhood and city, maybe even hide the fact that you’ve won in order to preserve relationships and a modest lifestyle – then you’re winning the lottery game (no purchase required). Although flying coach would definitely be a thing of the past.
I use an expectation of gain of about 0.7 as my buy or not-buy criterion. I nearly 20 years of playing the game, I’ve won one prize – which was a free ticket. Hardly worth the effort.
If you haven’t read the story of how Voltaire beat the lottery, it’s rather interesting: http://www.todayifoundout.com/index.php/2013/05/how-voiltaire-made-a-fortune-rigging-the-lottery/
One of the investigations that led Cardano to discover complex numbers was his effort to figure out how to gamble better.
I had a workmate who had a system for winning on the horses. He could only do this at racetracks where the ‘totalisator’ automatically calculates and displays the odds on a horse according to total bets received.
He calculated from statistical results that, while the favourite and outsiders were always ‘over-backed’, the third or fourth favourites were often ‘under-backed’ i.e. their chances of winning times the payoff offered were better than 1. I think he bet quinellas (i.e. first and second place).
He did quite well for a while till he had a run of bad luck and got wiped out.
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