The Foolishness of the Lottery

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I wondered -- could this be Foolish? In other words, what would the lottery prize have to be in order for it to make sense for me to invest a dollar?

For both of you who are unfamiliar with lotteries, here's how it works in Texas: go into nearly any convenience store. For $1 you get to pick six numbers between 1 and 50. If you don't feel up to the creative challenge, you can ask for a "quick pick" and the machinery will pick a random six numbers for you. If your numbers match the six drawn in the next drawing, you get to split a large jackpot with anyone else who happens to be right on them. Actually, if you match fewer numbers you can get smaller awards, but let's ignore that for now. We want to win the big one!

What are our odds of winning?

There are 50 possibilities for the first number, then 49 possibilities for the second number, then 48 possibilities for the third number, then 47 possibilities for the fourth number, then 46 possibilities for the fifth number, then 45 possibilities for the sixth number.

Now you get to find out how good your calculator is. If it doesn't fail on the calculation, you'll find that 50*49*48*47*46*45 = 11,441,304,000.

This enormous number, then, is all the possible sets of 6 numbers, in a specific order. However, in the lottery, the numbers can be in any order. For instance, if 1 2 3 4 5 6 wins so does 4 1 3 6 5 2 and 2 6 5 1 4 3 and many others. If you have 6 numbers, the number of possible orders for them can be counted in the same way as we counted the possible sets of winners. The first number can be one of the 6, the second can be one of the 5 left, and so on, so that 6 numbers can be in 6*5*4*3*2*1=720 possible orders.

So the number of possible winning numbers for any given lottery pick is 11,441,304,000/720 = 15,890,700.

Our odds of winning are one in about 16 million.

But, if we win the lottery, we don't get the stated amount (in the current case, $13 million). We get equal payments over 20 years which total $13 million. Or we can select the "cash" option and get half of the face value of the money up front. In other words, if the current value is $13 million, we'll get $6.5 million in real cash. That's easier to work with, and makes sense if you think you can get over 7.8% or so on your money. If I were to win the lotto, I'd pick the "cash" option.

Also, we have to pay taxes on our winnings. There are no state taxes here in Texas, but we have federal taxes of 40% (close enough). We get to keep about 60%. If you say more, don't forget you'll probably end up paying lawyer's fees in an effort to maintain a private life. Now, we figured out our odds are 1 in 15,890,700 of winning the lottery. So for our one dollar, we want to get at least $15,890,700 net (to have even odds) or better.

To have this much after taxes, we would want $15,890,700 / 0.60 = $26,484,500 before taxes. To get that in cash, we want the face value of the lottery to be more than twice that, or more than $52,969,000.

That's still too low--since I've ignored the chance I'll have to split it with someone else so far.

The lottery in Texas has very seldom gotten over $50 million. Based on the www.txlottery.org website, it appears that a prize over $50 million has only been awarded five times in 10 years. Only once was that prize of over $50 million awarded to a single person. The average number of people splitting these large prizes was over six. So I would want the prize to be over $300 million or more to consider playing.

It's never gotten that high.

So I don't expect to be investing in the lottery anytime soon.

**Related Link:**

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