Wednesday, April 18, 2012

Expected Payoff from Mega Millions

Lottery games usually advertise the enormous amount of money that can be won, and the luxurious life that such winning would bring. But everyone knows that the chance of winning the jackpot is miniscule. So in this battle of extremities, does the large amount of money or the small chance win out?

Information taken right from the Mega Millions website reveals that it takes $1 to play a game. The calculations put aside for each of these possible prizes, here are the monetary amounts of the prizes with their associated chances of winning:

Prize Chance ^ (-1)
Jackpot 175,711,536
$250,000 3,904,701
$10,000 689,065
$150 15,313
$150 13,781
$7 306
$10 844
$3 141
$2 75

The ultimate jackpot prize value varies consistently, but at present is estimated at $65 million. To get the expected payoff, simply multiply each earning by its likelihood, and sum the products. This can be easily done in Excel.

When the jackpot is $65 million, the expected payoff is only 55.18 cents. In fact, it takes slightly less than $144 million for the expected payoff to just reach $1. The jackpot recently reached record high of $656 million. With that, the expected payoff was slightly over $3.91. However, tax was not considered in any of these situation. Furthermore, as in the case of the recent record high, the winning amounts are split if more than one winner arises. In all, the small probability of winning seems to dwarf the large amounts of potential winning.

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