Mega Millions Lottery analysis // Lotteries

Mega Millions Lottery analysis

man with a pile of moneyThis could be you.Earlier this year, we considered the math behind deciding when to buy a Powerball ticket.

Now, the Mega Millions jackpot has reached a ludicrously high $636 million, and so we decided to repeat the analysis for this lottery— ahead of tonight's drawing is at 11 PM EST.

The most basic thing to consider in any game of chance, whether you are looking at a lottery, a roulette wheel, or a deck of cards in a blackjack game, is your expected winnings from the game. This is basically a measure of how much you will win on average by playing the game a large number of times.

The expected winnings for a game can be found if we know the probabilities of the different outcomes in the game, and the earnings (or losses) associated with each outcome. We multiply the probabilities by the earnings or losses, and then add these up.

The expected value will tell us what will happen on average if we play a game a large number of times. If the expected value is positive, then on average we are winning money, and so if we play the game long enough, we should end up in the black. If the expected value is negative, then on average we are losing money, and so we should not be playing this game.

We can find the expected value for playing Mega Millions with the current ridiculous jackpot. In Mega Millions, you pick five different numbers between 1 and 75, and one number between 1 and 15. If all six match, you win the jackpot. If you match some of the first five numbers and/or the last number, you win a smaller prize.

The Mega Millions website helpfully includes the odds of winning either the jackpot, or of winning various smaller prizes. Since the cost of a Mega Millions ticket is $1.00, we can subtract this from each of the prizes to get our actual take home winnings for each of these outcomes, and so find the expected value of a Mega Millions ticket:

Business Insider/ Mega Millions

The first column shows each outcome — how many of the five numbers from 1 to 75 we got right, and whether or not we got the last number from 1 to 15. The second column shows the prize and the third the prize less our investment of a dollar per ticket. In the fourth column, we have the odds of getting each outcome, as per the Mega Millions website. In the fifth column, we convert those odds into a probability between zero and one by taking the reciprocal of the odds. Finally, we multiply together our winnings less investment by the probability, and sum these up, giving us the expected value.

Notice that by far our most likely outcome is that we get none of the numbers right. There is about a 93% chance that we just wind up losing our dollar. Despite this, since the jackpot is so enormously high right now, our average winnings are a nice positive $1.63, indicating that we should consider buying a ticket.

Another notable property of Mega Millions is that the lower prizes do not help us too much. Without that jackpot, the expected value is a very unhappy —$0.82. So, Mega Millions really is all about the jackpot.

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Ware Man Cashes In $390000 Lottery Ticket Hours Before It Expires  — CBS Local
But that changed when Cote drove to Massachusetts Lottery headquarters in Braintree with his winning ticket, claiming a one-time cash payment of $390,000 before taxes.

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Popular Q&A

How does the California Mega Million Lottery work?

Find a store that sells lottery tickets.  In California your gas station or local 7-Eleven should sell these tickets.

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