Mega Millions lottery: Is it worth buying a ticket? We did the math

The current Mega Millions jackpot is the largest in
history.
AP Photo/G-Jun
Yam

• The Mega Millions jackpot is at a record-high $970
  million ahead of Friday’s drawing.
• Though that’s a pretty big prize, working through the
  math of how lotteries work suggests that buying a ticket is not
  a great investment.
• The low probability of winning and the risk of
  splitting the prize in a big, highly covered game mean you’d
  probably lose money.
• READ MORE:
  How to win the lottery, according to a mathematician who
  hacked the system and won 14 times.

The Mega Millions
jackpot is up to $970 million as of 9:45 a.m. ET on Friday ahead
of this evening’s drawing.

(The Powerball jackpot
is also up to a respectable, albeit much lower, $430 million
ahead of that game’s drawing on Saturday evening.)

That’s the
biggest Mega Millions prize ever, according to the lottery’s
website. However, taking a closer look at the underlying math of
the lottery shows that it’s probably a bad idea to buy a ticket.

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
expected
value.

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,
then,
in the long run, the game will make me money. If the 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 $2 ticket you buy, you choose
five numbers from 1 to 70 and one from 1 to 25. Prizes are based
on how many of the player’s chosen numbers match those drawn.

Match all six numbers, and you win the jackpot. After that, there
are smaller prizes for matching some subset of the numbers.

The Mega Millions website helpfully provides a list of the odds and
prizes for the game’s possible outcomes. We can use those
probabilities and prize sizes to evaluate the expected value of a
$2 ticket.

The expected value of a randomly decided process is found by
taking all the possible outcomes of the process, multiplying each
outcome by its probability, and adding all those numbers. This
gives us a long-run average value for our random process.

Take each prize, subtract the price of our ticket, multiply the
net return by the probability of winning, and add all those
values to get our expected value.

Business
Insider/Andy Kiersz, odds and prizes from Mega
Millions

We end up with an expected value of $1.45, which is positive and
above our breakeven point. That suggests it might make sense to
buy a ticket — but considering other aspects of the lottery makes
things go awry.

Annuity versus lump sum

Looking at just the headline prize is a vast oversimplification.

First, the $970 million jackpot is paid out as an annuity,
meaning that rather than getting the whole amount all at once,
it’s 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 $548 million.

If we take the lump sum, we end up seeing that the expected value
of a ticket drops all the way to $0.06, which while still just
above breakeven, is much less enticing than the headline figure.

Business
Insider/Andy Kiersz, odds and prizes from Mega
Millions

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 or exceeding inflation.

On the other hand, the state is investing the cash somewhat
conservatively, in a mix of US government and agency securities.
It’s quite possible, though 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, 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
money.

Factoring this in, if we’re taking home only half of our
potential prizes, our expected-value calculations move into
negative territory, suggesting that our Mega Millions investment
would be a bad idea.

Here’s what we get from taking the annuity, after factoring in
our back-of-the-envelope estimated 50% in taxes. The expected
value drops to -$0.15, below zero and therefore indicating that
buying a ticket is a losing proposition.

Business
Insider/Andy Kiersz, odds and prizes from Mega
Millions

The tax hit to the lump-sum prize is just as damaging.

Business
Insider/Andy Kiersz, odds and prizes from Mega
Millions

Even if you win, you might split the prize

Another problem is the possibility of multiple jackpot winners.

Bigger pots, especially those that draw significant media
coverage, tend to bring in more lottery-ticket customers. And
more people buying tickets means a greater chance that two or
more will choose the magic numbers, leading to the prize being
split equally among all winners.

It should be clear that this would be devastating to the expected
value of a ticket. Calculating expected values factoring in the
possibility of multiple winners is tricky, since this depends on
the number of tickets sold, which we won’t know until after the
drawing.

However, we saw the effect of cutting the jackpot in half when
considering the effect of taxes. Considering the possibility of
needing to do that again, buying a ticket is almost certainly a
losing proposition if there’s a good chance we’d need to split
the pot.

One thing we can calculate fairly easily is the probability of
multiple winners based on the number of tickets sold.

The number of jackpot winners in a lottery is a textbook example
of a binomial
distribution, a formula from basic probability theory. If we
repeat some probabilistic process some number of times, and each
repetition has some fixed probability of “success” as opposed to
“failure,” the binomial distribution tells us how likely we are
to have a particular number of successes.

In our case, the process is filling out a lottery ticket, the
number of repetitions is the number of tickets sold, and the
probability of success is the 1-in-302,575,350 chance of getting
a jackpot-winning ticket.

Using the binomial distribution, we can find the probability of
splitting the jackpot based on the number of tickets sold.

Business
Insider/Andy Kiersz, odds from Mega Millions

It’s worth noting that the binomial model for the number of
winners has an extra assumption: that lottery players are
choosing their numbers at random. Of course, not every player
will do this, and it’s possible some numbers are chosen more
frequently than others. If one of these more popular numbers
turns up at the next drawing, the odds of splitting the jackpot
will be slightly higher. Still, the above graph gives us at least
a good idea of the chances of a split jackpot.

Most Mega Millions drawings don’t have much risk of multiple
winners — the average drawing in 2018 so far sold about 19.2
million tickets, according to
our analysis of records from LottoReport.com, leaving only
about a 0.2% chance of a split pot. Even Tuesday’s drawing, which
brought in about 105.2 million tickets, according to
LottoReport.com, had only a 4.8% chance of a split pot, based on
the binomial-distribution analysis.

The risk of splitting prizes leads to a conundrum: Ever bigger
jackpots, which should lead to a better expected value of a
ticket, could have the unintended consequence of bringing in too
many new players, increasing the odds of a split jackpot and
damaging the value of a ticket.

To anyone still playing the lottery despite all this, good luck!