- The Mega Millions jackpot is now $US1.6 billion after Friday’s drawing with no winners.
- Though that’s a pretty big prize, working through the maths 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: Here are the winning numbers from the $US1 billion Mega Millions jackpot drawn October 19.
The Powerball jackpot was also up to a respectable, albeit much lower, $US620 million after its Saturday evening drawing.
That’s the biggest Mega Millions prize ever, according to the lottery’s website. However, taking a closer look at the underlying maths 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.
Expected values combine both probabilities and prize values to give us a better sense of the value of a game. As an example, consider a simple game where we flip a coin. If the coin is heads, you pay me $US2. If it’s tails, I pay you $US1.
Intuitively, this game seems like a bad idea for you, and expected value shows why.
To calculate the expected value, we multiply together the probability of each outcome by the value of each outcome and add them together. In our coin toss game, there’s a 1/2 chance you lose $US2, and a 1/2 chance you win $US1, so your expected value is (1/2 × -$US2) + (1/2 × $US1) which gives us -$US1 + $US0.50, for a final expected value of -$US0.50 for you.
Lotteries are a great example of this kind of probabilistic process. In Mega Millions, for each $US2 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 $US2 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.
We end up with an expected value of $US3.53, 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 $US1 billion 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 $US904 million.
If we take the lump sum, we end up seeing that the expected value of a ticket drops all the way to $US1.23, which while well above breakeven, is much less enticing than the headline figure.
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 generally 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 $US0.89.
But when we put everything together, looking at the more realistic choice of taking the lump sum, we end up with a negative expected value of -$US0.26 after taxes, suggesting that a ticket is a bad investment:
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.
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 21.5 million tickets, according to our analysis of records from LottoReport.com, leaving only about a 0.2% chance of a split pot.
But big prizes like the current Mega Millions game draw in many more players than usual. According to LottoReport.com, there were about 280.2 million tickets sold before Friday’s $US1 billion jackpot drawing. Following our binomial distribution model, in a drawing with that many tickets sold, there’s a 24% chance of a split jackpot, which would seriously damage the expected value of a ticket.
To see just how damaging the risk of a split jackpot is, we can estimate the expected value contingent on the number of tickets sold. As more tickets get sold, the risk of a split jackpot goes up, and we can include that in the expected value calculation:
When considering the risk of a split jackpot, the expected value for the pre-tax annuity falls below zero if around 860 million tickets are sold. The pre-tax lump sum becomes a losing bet if at least 360 million tickets are sold, the after-tax annuity has a below-zero expected value at around 270 million tickets, and as we saw above, the after-tax lump sum option always has a negative expected value, even before considering split jackpots.
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!
Read more of our Mega Millions coverage:
- No one has won the Mega Millions jackpot and it’s ballooned to $US1.6 billion – but if you do the maths, that’s not surprising at all
- The Mega Millions jackpot just hit $US1.6 billion – here’s how to play
- The record-high Mega Millions jackpot is worth $US1.6 billion – here are 3 things you should do if you win
- Buying enough Mega Millions tickets to cover every possible combination sounds like a surefire way to win – but there are 3 major problems with that plan
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