Friday’s Mega Millions drawing has a jackpot worth $393
A useful tool in deciding whether to buy a ticket is
the expected value of that ticket.
After factoring in taxes, it might be a bad idea to
spend your $1 on the lottery.
The Mega Millions drawing
for Friday evening has, as of noon ET, an estimated jackpot
of about $393 million.
While that’s a huge amount of money, buying a ticket is still
probably a losing proposition.
Consider the expected value
When trying to evaluate the outcome of a risky, probabilistic
event like the lottery, one of the first things to look at is
The expected value of a randomly decided process is found by
taking all of the possible outcomes of the process, multiplying
each outcome by its probability, and adding all of these numbers
up. This gives us a long-run average value for our random
Expected value is helpful for assessing gambling outcomes. If my
expected value for playing the game, based on the cost of playing
and the probabilities of winning different prizes, is positive,
in the long run, the game will make me money. If expected
value is negative, then this game is a net loser for me.
Lotteries are a great example of this kind of probabilistic
process. In Mega Millions, for
each $1 ticket you buy, you pick five numbers between 1 and 75,
and then an extra number between 1 and 15. Prizes are then given
out based on how many of the player’s numbers match the numbers
chosen in the drawing.
Match all five of the numbers between 1 and 75, and the extra
number between 1 and 15, and you win the jackpot. After that,
smaller prizes are given out for matching some subset of those
The Mega Millions website helpfully provides a list of the odds and
prizes for each of the possible outcomes. We can use those
probabilities and prize sizes to evaluate the expected value of a
$1 ticket. Take each prize, subtract the price of our ticket,
multiply the net return by the probability of winning, and add
all those values up to get our expected value:
Looking at the basic case, we end up with a positive expected
value of 69 cents, making it look like a Mega Millions ticket
might be a good investment.
But there are a few catches.
Annuity versus lump sum
Looking at just the headline prize is a vast oversimplification.
First, the headline $393 million grand prize is paid out as an
annuity, meaning that rather than getting the whole amount all at
once, you get the $393 million spread out in smaller — but still
multimillion-dollar — annual payments over 30 years. If you
choose instead to take the entire cash prize at one time, you get
much less money up front: The cash payout value at the time of
writing is $246 million.
Looking at the lump sum, we get a pretty big cut into our
expected value, which falls to 13 cents:
The question of whether to take the annuity or the cash is
somewhat nuanced. The
Mega Millions website says the annuity option’s payments
increase by 5% each year, presumably keeping up with and somewhat
On the other hand, the state is investing the cash somewhat
conservatively, in a mix of various US government and agency
securities. It’s quite possible, although risky, to get a larger
return on the cash sum if it’s invested wisely.
Further, having more money today is frequently better than taking
in money over a long period of time, since a larger investment
today will accumulate compound interest more quickly than smaller
investments made over time. This is referred to as the time
value of money.
Taxes make things much worse
In addition to comparing the annuity with the lump sum, there’s
also the big caveat of taxes. While state income taxes vary, it’s
possible that combined state, federal, and, in some
jurisdictions, local taxes could take as much as half of the
Factoring this in, if we’re taking home only half of our
potential prizes, our expected-value calculations move into
negative territory, making our Mega Millions “investment” a bad
idea. Here’s what we get from taking the annuity, after factoring
in our estimated 50% in taxes. The new expected value is now
underwater, at -7 cents:
The hit to taking the one-time lump sum prize is just as
After factoring in taxes, then, our “investment” in a Mega
Millions ticket doesn’t necessarily seem like a great idea.