Was Playing Megamillions Rational?
The multistate lottery “Megamillions” had a single winner in yesterday’s drawing. Because of an unusually high number of drawings with no winner, the jackpot had grown to approximately $250 million.
Which suggests that it might have been irrational not to play the lottery this time around.
Without going into the specifics of this lotto-style game, and with no lecture on my favorite branch of mathematics, combinatorics, there are (trust us) 175,711,536 possible combinations for the Megamillions game.
It costs $1 per combination to play the game. Hence the expected value of a $250 million lottery ticket is
$250,000,000 x 1/175,711,536
or
$1 x 250,000,000/175,711,536.
Since 250,000,000/175,711,536 is greater than 1, the ticket was, probabilistically speaking, worth more than $1 (in fact it was worth $1.42). Yet the ticket only cost $1.
It was therefore, at first pass, irrational not to buy lottery tickets.
Of course, in the real world the fact that the jackpot is subject to taxes, and is structured as an annuity and must therefore be discounted, makes the ticket worth far less than $1.42, and probably less than $1. In addition, there is always the risk that someone else will also choose the correct numbers, in which case the jackpot is shared evenly. (On the other hand, the fact that there are “runner-up prizes” would make the ticket worth even more than $1.42 pre-tax, pre-discounted).
Still, it’s interesting to see a lottery swell to such levels that the irrationality of playing it shrinks to nearly zero. Does the Law of Large Numbers require that, eventually, a jackpot would grow so large as to be a rational bet? (I think not, since people change their behavior as the jackpot grows.)
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Megamillions is a lotto-type game: pick so many numbers between 1 and whatever. The odds of picking correctly are 1 in 175,711,536. Lots of people play Megamillions. But consider an alternative lottery, with the same jackpot structure but played as follows: Simply pick a number between 1 and 175,711,536. Would anybody play that game? Aren’t people who play Megamillions but who wouldn’t play the “pick a number” game irrational? Why might they not be? (Hint: See here.)
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The number where the expected return (with all the runner-up prizes included) equals the cost of the ticket is around $100 million. (I'd like to see this number adjusted for the tax and lump-sum conversion).
My favorite lottery "fact" is that playing more is significantly wiser than playing just 1 ticket. Why settle for 1 in 175,711,536 when you can get 1 in 8,785,577 just by buying 20 tickets. I've tried to correct people on this in the past, but it never works. The lottery is most definitely a tax on people who are bad at math.
[Kip replies: So does it make sense to bet on every horse in the race, since you're guaranteed to win?]