Pseudo-Random Numbers

Ever stop and wonder what that little $1 ticket is worth before it becomes worthless? How much intrinsic value does a near-impossible chance at near-unthinkable riches really have?

Well, to your run-of-the-mill economist, a 10% chance at $100 is worth $10. Prediction markets are all over the place that work on the exact same principle. I decided that since the odds are printed right there on the ticket, there ought to be a way to figure out what that value really is.

Where the math gets tricky is when you add in the possibility of splitting a jackpot. If yours was the only ticket, and the jackpot (after taxes) was 100 million dollars, and the odds of winning were 1:100,000,000 then the ticket would be worth one dollar, easy. But what if there are 100 million other tickets out there? All of a sudden, you’re in a position where, even if you hit that one in 100 million winning ticket, you’ve only got a roughly 32% chance of keeping all the money to yourself. Obviously the ticket is no longer worth one dollar, but it’s still worth something.

The formula I came up with goes something like this:

where 1/P is the probability that any single ticket will match the winning numbers, k is the total number of tickets in play, and J is the jackpot (after taxes, of course).

However, when P and k are both very large (as they are for any lotto drawing), we can make the following substitution:

in this case, ‘a’ is the probability that a single winning ticket won’t have to share the jackpot with any other winning tickets.

Using ‘a’ as our substitution variable in the previous equation, and assuming that, in the real world, it’s downright silly to imagine more than 10 tickets all winning the jackpot, the following approximation takes shape:

Now, it may not get you every last femto-cent, but it’ll get you within a tenth of a cent sure enough, and that’s as far as I’m interested in.

Of course, this equation still leads to some head-scratching, because it’s hard to apply as-is. You might know the cash value of the jackpot, but you probably don’t know how many tickets are in play. Luckily, computers can do some amazing sh*t. There are 2 major multi-state drawing-type lotteries in the US - Mega Millions and Powerball.

Each of the drawing history pages shows the number of winning tickets at every prize level. I know that the odds of winning any prize in Mega Millions is about 1 in 40, and the odds of winning any prize in Powerball is 1 in 36.61. Given that, plus knowing the total number of winners each week, I can create a scatter plot using Excel to see the correlation between the advertised jackpot and the number of tickets sold for that drawing.

I discovered an interesting fact while plotting these numbers - lottery ticket sales are trending up. The same jackpot today will tend to sell more tickets than it would a year or two ago. However, when I only plot data points that are less than 18 months old, I can spit out a classic quadratic function with an R-squared value of .9955 for Mega Millions, and another quadratic for Powerball with an R-squared value of .9875.

Here they are:

1. Mega-Millions:
2. Powerball:

In this case, k is the number of lottery tickets in millions, and J is the advertised jackpot in millions. By looking at it, you can see that no matter what the advertised jackpot, Mega Millions can count on selling at least 17 million tickets and Powerball 14 and a half million.

Now, a couple things to notice. This equation is only good with real-world values up to about the 300+ million, since those were the highest jackpots ever recorded, as well as my highest data points. I imagine at some point the graphs would flatten out, because no matter how high the jackpot gets, there has to be a limit to how much people can spend on lottery tickets. Where exactly the equation would deviate from real values I’m not sure, but don’t imagine you can go plugging in a 1 billion dollar jackpot and expect you can trust the answer it spits out. The second thing to keep in mind is that, as I mentioned before, the equation changes over time. The values from 2005 are all way too low to plot on this curve, just as I imagine the values in 2011 could be well off too. Spending habits change, and they’re reflected here. I don’t know if people will still reference this blog entry five, ten or twenty years from now, but if they do, they need to recreate the equation based on fresh data.

Ok, ok, so that’s all well and good, but do you really expect me to do all this math just to figure out the value of a stinking one dollar ticket?

Of course not. All these functions can easily be plugged into a simple computer program, and that’s exactly what I’ve done for you. In addition to the main jackpot, you can add in the lesser prizes. After all, a 1:50 chance of winning $2 is worth 4 cents. That math is pretty easy, since it isn’t affected by the number of tickets in play (except in California for some reason).

My solution was to build an ajax-powered calculator. Go check it out on my PHP portfolio blog. To make it cooler, it will automatically fill in the information for the Powerball and Mega Millions lotteries for you. If you just click the link to set it up for either drawing, all the odds and prizes will show up, along with the current jackpot’s cash value and approximate number of tickets in play. Just set it up and hit “calculate” to immediately discover the value of one random lottery ticket.

Check out the calculator here:
http://hiremasonwolf.com/calculating-the-value-of-a-single-lottery-ticket/58

Have fun, but don’t expect to ever want to play the lottery again!

Filed under: probabilities