Probability and MegaMillions

The comments on this infographic inspired some LottoMillions thinking. What is the probability that you win the jackpot and no one else wins?

http://www.popsci.com/science/article/2012-11/dismal-odds-winning-lottery-infographic

The odds of winning are based on you selecting the correct numbers out of a larger set of numbers. Keep in mind that the odds of being a lottery winner (not the same thing as winning the lottery) are # of people who have won divided by all people or divided by all the people who have played the lottery – depending on how the author frames the probability.

But what about odds of wining the big jackpot and being the ONLY winner??

Typically the odds reported are based on simple selection. For example, the Mega Millions is played by selecting 5 numbers between 1 and 56 in the first five spots and a single number from 1 to 46 in the sixth spot, the odds of choosing the winning combination would be 1 in 175,711,536. This is the number most outlets report, approximately 1 in 176 Million.

But what are the odds the you make the correct selection and no one else does? Do your odds get better or worse?

prediction_graphs

Let’s Warm Up

Imagine you can win a prize by selecting the one correct letter out of 5 letters: A, B, C, D, E. The probability of choosing the correct letter is 1/5. Those are your odds for winning. But those are the same odds for winning of everyone playing. The more people who play, the smaller your jackpot is likely to be. And you don’t like sharing.

So what are the odds of you winning and no one else winning?

You winning: 1/5
Not winning: 4/5
No one else winning:  (4/5)^(number people who played)

For you to win the prize and not need to share, you must win and everyone else playing must lose. Imagine you win and I lose:

1/5 * 4/5 = 4/25

LotteryThe probability of you winning was 20%, but the probability of you winning AND me losing is only 16%. Does that change as the number of players increases?

You, Will, and Sarah are playing. You win and Will and Sarah lose:

1/5 (you win) * 4/5 (will loses) * 4/5 (sarah loses) = 16/125 or 12.8%

The group multiplier is (4/5)*(4/5) or (4/5)^ (number of people who must lose). If 10 people are playing, the probability of you winning and everyone else losing is  . . .

1/5 * (4/5)^9 = .0268 or 2.68%

So even with a small group of additional players the chances of you winning and everyone else losing drops dramatically. For every additional player, your chance to snag the prize without sharing drops by 20%, the same percentage as any person’s chance to win. You have an incentive to keep people out of the game.

With respect the to Mega Millions, your chance to win is 1 in 175,711,536 or .00000000569 or .000000569%. Your chance to win and not share the prize if 3 people are playing: .00000000569, or .000000569%

you win: 0.0000000056911459701

you win and the other two lose: 0.0000000056911459053

The probability of winning and everyone else losing does not change much as you add more people to the mix because the probability of losing is so close to 1 or 100%, that raising the loser multiplier (.999999994) to a power doesn’t change the group multiplier as much. In other words, because each person’s chance to win is so small, your incremental opportunity to win does not decrease much as more people play. For the MegaMillions you don’t have much incentive to keep people out of the game. Unless . . .

you rig the game . . .

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