The criminal the luminary Rev. Dr. King has a lot to teach us, including to avoid the worst argument in the world.
I was heartbroken to see this rampant racial segregation in America, on MLK Day no less!
I was informed by WordPress that some of the ad revenue on Putanumonit will begin to accrue to yours truly. I assume that beforehand your monetized clicks were simply drifting into the dark abyss. I reiterate that the financial goal of this blog is strictly to lose money, I will
drink away donate to charity whatever ad-pennies accumulate.
Here’s the full, easy to browse archive of Putanumonit. I’m trying to get y’all to use the comments there for topic suggestions, so far to no avail.
Remember how the tails of a bell curve drop off much faster than you ever imagined (and that’s why China is bad at soccer)? Even Francis Galton, a founding father of statistics and sampling theory didn’t fully grasp it.
SlateStarCodex breaks down another abuse of statistics by the media, this time by a right-wing source to show political neutrality. Since I care more about meta politics than about politics I don’t worry about the latter, and I also assume a priori that any article on a political site is probably using numbers to bullshit. If I ever come across a political article that uses sound, well-interpreted data to make an unbiased analysis that would be newsworthy enough to write about. I’ll try to go after tougher targets: published scientific research and those in the media who should know better.
Speaking of 538, Walt Hickey wrote a couple of pieces on estimating the number of Powerball tickets sold. His original model (which made perfect sense given the data at the time) was of exponential growth, where doubling the jackpot would more than double the number of tickets. Faster than linear growth would mean that there would be an optimum jackpot beyond which each participant’s winnings will decrease as the prize increases.
However, the two recent mega-jackpots don’t bear out that case, with Wednesday’s Powerball topping out at 635 million tickets. That’s 3 tickets per adult in the United States, but still below exponential. I fit a linear regression model of the number of tickets sold based on the size of the jackpot and the number of news articles about Powerball for each drawing.
I have included all data since Powerball switched to $2 tickets in January 2012. Data from 1/2012-4/2013 are adjusted for the fact that California joined Powerball on 4/10/2013 and has since accounted for almost 11% of ticket sales.
The model gives a baseline of 7.5 million tickets for a minimal $40m prize (actual minimum is around 10 million) and 170,000 extra tickets for every $1m in the pot. Every news article (which tend to materialize when the jackpot is the “biggest ever”) inspires 5 million tickets, or one extra winner per 57 news stories.
Based on these numbers, can a jackpot grow large enough to provide a positive expectancy on your money? Before I answer that, there’s another important point to examine.
In a lottery with 600 million tickets sold and 1 in 300 million odds of winning, we expect to have two winners on average. Imagine discovering that you have won the lottery, how many people do you expect to share the prize with? Many people can’t get over the intuition that there should be one more winner besides them, but that’s not the case: given that you won you should expect two additional winners!
The math is straightforward assuming that the tickets are independently generated: finding out that you have won gives you no information about the other 599,999,999 tickets. Each ticket has 1 in 300 million chances of winning so two of them (1.999.. if you’re nitpicking) will win, on average.
Here’s another way to look at it: before finding out that you won, you didn’t know how many winners there were going to be. For all you knew, there could be 0, 2 or even 10. Once you know that you have won, you know for certain that there weren’t 0 winners – there’s at least you. In fact, knowing that you won makes worlds with more winners relatively more likely. This is because the more winners there are the more likely you are to find yourself among them! Such is the awesome powa of Bayes’ Theorem. Here’s a good detailed explanation of the theorem. The theorem itself is only useful for inconsequential things like learning anything at all based on evidence.
The charts above show the probability of Wednesday’s Powerball (635 million tickets, 1 in 292 million odds) having a certain number winners. On the left is the number of winners we could expect before the drawing: 2.17 on average. On the right is the conditional probability of having k winners given that you won, and it indeed averages to 3.17. Isn’t the discrepancy paradoxical? While each of the (few) winners will adjust their estimate of the number of winners upward from 2.17 to 3.17, each of the (numerous) losers can perform a similar calculation and adjust the expected number of winners slightly downward, to 2.169999. A small chance of updating strongly upward is balance by a huge chance of updating very slightly downward. Expected evidence is conserved, the paradox is avoided, and if you win you have to share.
So what would it take for a positive value Powerball? It would take a bit of algebra!
A $2 ticket grants a 1 in 292,201,338 chance of winning, so the expected first prize should be
Unfortunately, even without state tax you would pay federal income taxes of 39.6% which means that your expected winnings will have to total
You’ll expect to share the prize, so
Assuming all tickets are independent:
Finally, we’ll use my linear model for the expected number of tickets.
The two mega-jackpots this month generated at least 100 news items (as counted by Google) between them, so we’ll assume 50 news stories for our presumed titanojackpot. This means a jackpot of $4.1 billion, which would provide break even value for each of the 950 million tickets sold!
Don’t hold your breath. Assuming that the government reinvests 80% of revenue (it doesn’t) and that it won’t change the rules along the way (it will), the chances of a run of jackpots going from $40 million to $4.1 billion with no winner along the way are 1 in 160,000. That means that we can expect the first positive-value Powerball to happen in the year 15,692 AD! And when we get there, we’ll find that some hedge fundies have bought all the tickets beforehand.