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January 30, 2023

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# How To Calculate Odds Of Winning \$1.1 Billion And The Value Of A Mega Millions Ticket

Looks like each \$2 ticket is worth \$3.60. But it isn’t.

Likely jackpot for the July 29 Mega Millions drawing: \$1.1 billion. Odds your ticket will win it: a bit longer than 1 in 300 million.

Divide one number by the other. It seems the expected payoff of a \$2 ticket is upwards of \$3.60. So, you should buy an armload of them?

No, and this essay will walk through the arithmetic of why. If you must, buy one ticket, for fun. Buying two would be foolish.

There are three reasons why the jackpot is too meager to give your ticket a positive expected return. The first is that the advertised number is misleading. The pot is to be disbursed over 30 years. If you want it all at once, it shrinks to \$648 million.

Already, your expected payoff has dwindled to \$2 and change, scarcely better than a breakeven.

Next problem is that you might be sharing this money. There’s a 35% chance there will be one other winner, so you’ll get only half. There’s a 40% chance you’ll be sharing with two or more other winners.

The risk of having to share cuts the jackpot value down to \$1.17.

The last haircut is for taxes. The federal rate on a big windfall is 37%. The state and local rate could be anything from 0% (such as in Texas) to 14.8% (in New York City). The payoff table below assumes a 5% state and local hit.

With a combined 42% tax, your expected payoff from the jackpot goes down to 68 cents. The smaller prizes are worth a little something, but not enough to push the value of a lottery ticket above a dollar.

There is a select group of lottery players who do slightly better because their take from the smallest payoffs is exempt from tax. That group is people who are both rich and stupid. I’ll explain later.

The Mega Millions game consists of picking five different numbers in the range of 1 to 70, and one Mega number in the range of 1 to 25. The first five are like a poker hand and their order does not matter. The Mega number can be the same as one drawn in the first group.

To win the jackpot, you have to hit all six numbers. If you match the first five but not the Mega ball you get \$1 million. There are smaller prizes for lesser matches.

I’m leaving out of this discussion a “multiplier” option that costs an extra dollar and gives you better payouts from the sub-jackpot prizes.

The small prizes are fixed. The jackpot isn’t. It starts out small and grows every time there’s a drawing with no jackpot winner.

At the end of the July 26 drawing, the pot was \$830 million. The lottery operators estimate that a bit more than 415 million tickets will be sold for the July 27 drawing, for \$831 million in gross sales. A 32.5% piece of that, or \$270 million, goes into the pot. That would yield a Friday night pot of \$1.1 billion. That’s before the discounting for an immediate payout.

Don’t pray for a last-day buying frenzy that makes the jackpot bigger than the official estimate. It would also increase the hazard of other winners horning in on your money. The net effect of ticket sales at this point is to lower the value of each ticket.

Here’s my calculation of what a ticket is worth:

You may be wondering where some of these numbers are coming from.

To get the odds on a jackpot win, start by multiplying out this quantity:

70 x 69 x 68 x 67 x 66,

which is the number of ways to draw five balls from an urn of 70 balls, without replacement, and then divide by this quantity:

5 x 4 x 3 x 2,

which is the number of ways to sequence five objects (remember, the lottery doesn’t care about order).

When you’re done, multiply by 25, because you’ve got only a 1 in 25 chance of nailing the Mega ball.

Result: 1 chance in 302,575,350 of winning. Or, in other words, odds of 302,575,349 to 1 against.

A match of four of the main numbers is 325 times as likely. That’s because there are five different numbers to get wrong, and, for each, 65 different wrong numbers.

As for sharing, that gets a little complicated. You need to calculate a sharing factor. Add the probability that there are no other winners…to ½ times the probability there’s one other…to 1/3 times the probability there are two others…and so on.

I’ll skip the detail but note that the first of these probabilities (that none of the other 415 million tickets is a jackpot winner) is very closely approximated by e^-R, where e is a number you were supposed to learn in math class and R is the ratio of 415 million to 302.5 million.

What about avoiding the tax bill? That’s hard to do with the larger prizes. For the small payouts, you can take advantage of the tax deduction for gambling losses. If you are a regular player and keep your losing ticket stubs, you might have enough to write off a \$500 win. However, the deduction is limited to the amount of winnings you report, and it can be claimed only by people who itemize deductions.

Generally speaking, in order to get a benefit from itemizing you have to be rich. In order to be a regular lottery player you have to be stupid.