You've seen the articles that say "MCMC is easy! Read this!" and by the end of the article you're still left scratching your head. Maybe after reading that article you get what MCMC is doing... but you're still left scratching your head. "Why?"
"Why do I need to do MCMC to do Bayes?"
"Why are there so many different types of MCMC?"
"Why does MCMC take so long?"
Why, why, why, why, etc. why.
Here's the point: You don't need MCMC to do Bayes. We're going to do Bayesian modeling in a very naive way and it will lead us naturally to want to do MCMC. We'll understand the motivation and then! We'll also better understand how to work with the different MCMC frameworks (PyMC3, Emcee, etc) much better because we'll understand where they're coming from.
We'll assume that you have a solid enough background in probability distributions and mathematical modeling that you can Google and teach yourself what you don't understand in this post.
- With the latest odds and given the way ticket purchases grow with the expected jackpot, the expected value of Powerball is negative. Even more so with a billion dollar payout.
- There is a 95% chance that there will be 3 or more winning tickets after the next draw. Almost zero chance no one will win.
- Don't laugh at the premise! Powerball did have a positive expected value when the expected jackpot fell within a range of $400m-$650m. The game was changed to reduce the odds in October, 2015 which "fixed" that problem.
First of all big shout out to Walter Hickey at Business Insider for the pointer to the Powerball data (here:http://www.lottoreport.com/powerballsales.htm) in this post a few years ago (here: http://www.businessinsider.com/heres-when-math-says-you-should-start-to-care-about-powerball-2013-9).
This chart plots the relationship between expected value of purchasing a ticket to estimated size of a jackpot. The model used takes into account the dramatic increase in tickets purchased as the jackpot size increases.
My analysis shows that because of the exponential increase in the number of tickets being played and the likewise dramatic increase in the likelihood of sharing the winnings, there is never a point where one will break even on buying a Powerball ticket... Now that they've made it harder to win.
Previous to 10/07/2015 this is what the expected value looked like:
Notice anything funny? Yeah it used to have a range of values where the expected value was positive! That means if the expected jackpot was somewhere between $400m-$700m it was actually a real investment for you to play the lotto. When the Powerball odds were reduced though, this stopped being a problem for the lottery. (You can read more about it here: http://news.lotteryhub.com/powerball-odds-set-change-october-2015/). Depending on how someone plays, the expected value may not mean much. Specifically for people who play only a few times in their lives since they won't play enough for the Law of Averages to even out the times when a player lost. A lot. Even so, I use it here because it's a pretty simple and intuitive framework to use to understand the value of an investment.
Let's dig deeper into the game with the current (and harder to win) odds to understand why, even in the face of a billion dollar payout, Powerball is a net negative play with these new odds.