Wednesday, November 30, 2011

The Interminable Statistical Tail of Plastic Fruit: A Monte Carlo Analysis of Hi Ho Cherry O!

I remember playing and enjoying the game Hi Ho Cherry O! as a kid. It is just the kind of unskilled experiment where individuals vest arbitrarily in the outcome of a random number generator that children[1] enjoy. But I recently oversaw a game between two three-year-olds that went on, and on, and on. The toddlers patiently awaited the outcome, spinning and obeying the spinner with surprising, unquestioning[2], attention. But I was losing my mind. So I occupied it by wondering if the game was statistically stable. It was a question that occupied me well past the termination of the interminable match. So I wrote a Monte Carlo analysis to answer it.

In case you are unfamiliar, here are the terms of this particular random number generator: The object of the exercise is for one party to achieve a score of +10.[3] There are seven equally likely outcomes for each random event, including four positive outcomes, a +4, a +3, a +2 and a +1 event. On the negative side, there are two -2 events.[4] So far it seems like a problem that would rapidly converge to a +10 solution. However, it is the seventh outcome (the third negative outcome) that adds the dramatic non-linearity. The ‘bucket spill’ event resets a player’s score to zero, meaning that it can register anywhere between 0 and -9…counteracting the convergent positive bias of the event.

The Results: The ‘bucket spill’ event does, in fact, produce a highly skewed distribution of play time. In 50,000 simulated games the average game lasted 9 turns (18 spins) but the longest game lasted 71 turns (142 spins) and there is a non-trivial probability (~5%) of a game that lasts more than 20 turns (40 spins).

Now, aren’t you glad I’m blogging again? I mean, seriously, where would the internet be without this kind of essential analysis?[5]

This post was finished[6] while listening to The Recluse by Cursive

[1] And adults, if the popularity of fantasy football is any indication…or the ability of adults to vest in one of those ’racing dots’ on the jumbotron at major sporting events. This, incidentally, is why the ‘sausage race’ (which is as near to literal and as far from ‘disturbing euphemism’ as it possibly could be) is one of the great intermission events in all of sports. The outcome is uncertain and anything can happen (including assault by a player). It is easier to vest and cheer with gusto for the Italian Sausage or Bratwurst. There is an insight here about the theology of election or philosophical determinism or openness theology, but I have committed to make this an entirely un-philosophical post, so I refuse to explore it. (Though, actually, I find the philosophy of time practically impenetrable, so really I am just punting. It’s a blog. I am satisfied to make the connection between sausage racing and election – I don’t necessarily need to analyze it.)
[2] Besides a few attempts at “Subtle Cheating.”
[3] The actual length of the game is driven by the difficulty toddlers have in hanging little plastic cherries on cardboard trees. I suspect the pedagogical value is as much in fine motor skills as it is in counting.
[4] Apparently birds and dogs both consume cherries in pairs. We had cherry trees when I was a kid and I can report that this seriously undervalues the consumptive potential of birds and overvalues canine appetite for fruit. But, back to the statistics…
[5] Seriously, though, here is the short term plan. I will run short pieces (mostly of more import than this one), a Fragments and Links, and my 2011 books post through the end of the year and then will do a 2-5 part series on campus dating and sexuality (including another nerdy statistical post, some thoughts on Donna Freitas’ Sex and the Soul, and hopefully reflections on Beth Bailey’s From the Front Porch to the Back Seat: A History of Dating – the two most scholarly yet helpful texts I have found on the topic) in January leading up a talk I am going to give on the topic.
[6] I am actually listening to a play list that is composed of ‘bookmarked’ songs from Pandora that I finally just bought.


Joel said...

I feel kind of like you should go ahead and oversee 50,000 games, just to make sure your stats are correct.

Or, at least, that you should also perform an in-depth analysis of "Candyland", too.

JMBower said...

I'm fairly sure if you play enough games of hungry hungry hippo, you have a statistically significant chance of splitting an atom, given the force we put into it as kids.

That being said, your ability to come up with this idea, run this analysis, and post about it while managing the other pressing concerns of your life make once again even more certain you have a vast warehouse full of top men working on these things for your benefit. Any other explanation would have to assume a superhuman level of time management:)

Aeiou1962 said...

This kind of thinking fascinates and frustrates me to no end. There is understanding and access to a whole new world view in it, and I can't get there from here. It's another language. I'm one of those people who took some satisfaction in saying that I've never once needed algebra since graduation (actually I once used pi to make brownies, because all I had were round cake pans and the recipe called for square). The only numerical pursuit I had in college was taking statistics in grad school, which accounts for what I think is a vague understanding of what is going on here. I just don't speak math, and I am reminded here, once again, that I am the poorer for it.