Financial Mathematics Text

Saturday, May 18, 2013

Is the Powerball lottery a good value?

The Powerball lottery jackpot for Saturday May 13, 2013 is estimated to be around \$600 million. That raises the question, with it being this high, is it a good gamble?

If math bores you, I'll give you a hint (it's not.) To show this is the case let's take a look at the payouts and the odds as listed here.

powerballl odds

In the third column you'll see the expected value associated with the non-jackpot portion of the winnings. In the final column, I subtract off 35% for taxes which will be relevant (especially for a jackpot winning.)

But \$0.23 isn't the entire thing. The largest portion of expected value is of course the jackpot. So it really depends upon how big it is. But there's a few problems.

  1. The stated jackpot size is not the present value. What they're actually doing is summing up the undiscounted future cash payments of the annuity.
  2. If you win a secondary prize, you get the prize as stated. But if you win the jackpot, and someone else also has the winning numbers, you have to split the jackpot with other winners.
If you check the Powerball website,  it mentions in fine print what the present value of the jackpot is worth. It's not the stated \$600 million but only \$376.9 million (surely nothing to complain about.)

Then we'll also want to discount that amount for taxes that need to be paid. Again, I'll be assuming 35%.

Lastly, we'll ignore the second problem above and assume, for simplicity, that you win the full amount and do not have to split it with one or more other winners. So here's what the total expected value looks like:

Powerball Expected Value

Keep in mind these tickets cost \$2 a piece. So the jackpot will need to reach almost \$760 million before you'll break even. And that's not factoring in the fact that you may end up having to split the jackpot.

So what are the odds you'd end up splitting the jackpot? That all depends upon how many tickets are sold. According to information from this site, there were over 90 million tickets sold on the drawing for May 15, 2013. If we round that up to a nice 100 million, there's over an 11% chance that the prize will be shared by 2 or more individuals. And the number of ticket sales seems to be correlated with the size of the jackpot.

So it's not entirely clear that the Powerball lottery is ever worth playing. It is a tax on people that can't do math. But I suppose it's some fun entertainment.

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