Big Money

Some say lotteries are a tax on people who are bad at math. But how big a jackpot would make buying a lottery ticket a good investment?

Last week, two people split $363 million, the largest lottery jackpot ever awarded in the United States.

In this lottery, the Big Game, 5 balls are drawn from a set numbered 1–50, plus a Big Money Ball from a set numbered 1–36. The number of combinations of jackpot numbers are are ((50×49×48×47×46) / (5×4×3×2×1)) × 36, so your odds are 1 in 76,275,360.

[calculation in MathML:]

$\left(\begin{array}{c}50\\ 5\end{array}\right)\times 36=\frac{50!}{5!\left(50-5\right)!}\times 36=\frac{50\times 49\times 48\times 47\times 46}{5\times 4\times 3\times 2\times 1}\times 36=76275360$

Lottery tickets cost $1, but their actual value is a sum of each prize times your odds of winning it:

Prize	Pick	Odds		Value
Total				$0.1799	+ x / 76,275,360
$1	0+MB	1 :	62	$0.0161
$2	1+MB	1 :	102	$0.0196
$5	2+MB	1 :	528	$0.0095
$5	3	1 :	220	$0.0227
$100	3+MB	1 :	7,705	$0.0130
$150	4	1 :	9,686	$0.0155
$5,000	4+MB	1 :	339,002	$0.0147
$150,000	5	1 :	2,179,296	$0.0688
x	5+MB	1 :	76,275,360	x / 76,275,360

For the value to equal $1, x = $62,545,795.20. However, to actually keep $62 million of your winnings, the jackpot must be more.

Estimating federal and state tax on the jackpot at around 40%, multiply by 1.4 to make approximately $87,500,000. (I ought to figure the tax on the $150,000 prize too, but I’ll let it go. It’s the Big Money we’re interested in, right?)

We’ll choose a lump sum payout rather than a 6% annuity, since we can get a much better return by investing. I estimate the lump sum payout alternative at half the annuity face value. (50% is good enough for an estimate. A winner of $180 million is taking $90 million cash, and the Pennsylvania lottery advertises prizes like $16M annuity or $8M cash.) Thus, to receive $87,500,000 cash, the jackpot must be a $175,000,000 annuity.

Unless more than one person wins. In this case, two people split a jackpot. If the average number of jackpot winners is two, that doubles the jackpot necessary for a ticket to be worth $1, to $350,000,000. (However, if the average number of jackpot winners were more, the jackpot would have to be too.)

Therefore, this $363 million lottery may have been the first lottery in US history in which a ticket was a good investment.