# What is the relationship between the Poisson Distribution and the Monte Carlo Fallacy?

Gravity's Rainbow has this long passage about the Poisson distribution. Since Pynchon's education included a serious dose of mathematics, and his novels include many references to mathematics, I assume what the characters are saying to each other must make some sort of sense, i.e. must have a formulation in mathematical language. But what exactly are they describing? What is the Monte Carlo Fallacy, and what does it have to do with the Poisson Distribution?

The two characters are looking at a grid which represents London. Places where bombs have hit are marked on the grid. Further up in the dialogue, the grid is compared to a sieve the Romans would have used for fortune-telling.

"Can't you . . . tell," Pointsman offering Mexico one of his Kyprinos Orients, which he guards in secret fag fobs sewn inside all his lab coats, "from your map here, which places would be safest to go into, safest from attack?"

"No."

"But surely!"

"Every square is just as likely to get hit again. The hits aren't clustering. Mean density is constant." Nothing on the map to the contrary. Only a classical Poisson distribution, quietly neatly sifting among the squares exactly as it should . . . growing to its predicted shape. . . .

"I'm sorry. That's the Monte Carlo Fallacy. No matter how many have fallen inside a particular square, the odds remain the same as they always were. Each hit is independent of all the others. Bombs are not dogs. No link. No memory. No conditioning."

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I really need to get around to reading Gravity's Rainbow... –  David Mitra Jan 4 '12 at 18:14
i used festival tts and it worked so well that i know what's on pretty much every page. also wrote about it here... couldn't focus on it without using audio though. literature.stackexchange.com/a/1135/355 –  ixtmixilix Jan 4 '12 at 22:42

The Monte Carlo Fallacy is more commonly known as the Gambler's Fallacy.

The fallacy is to believe that in a a series of independent events the outcome of the next event depends on the outcomes of past events. For example, a gambler might believe that the next spin of a roulette wheel is more likely to come up red if it has just come up black six times in a row. However, this isn't true -- that is precisely what it means to be independent.

However, note that it is not always a fallacy to believe this! For example, when playing blackjack, if many aces have just been dealt then it will be less likely that an ace comes up on subsequent draws (until the deck is reshuffled, that is). The reason is that deals from a pack of cards are not independent.

This isn't connected to the Poisson distribution per se. Rather, the Poisson distribution enters because it is related to the uniform distribution. If events (e.g. bombs being dropped) are distributed uniformly at random across a region of space, then the number of events occuring inside a particular fixed area will follow a Poisson distribution. For more information see the Wikipedia articles here and here.

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I'd say that the relation to Poisson is that Poisson distributions calculate accumulations of independent events. The independence is the important thing, in both Poisson and the Monte Carlo fallacy. –  Thomas Andrews Jan 4 '12 at 18:25
This is somewhat similar to flipping a coin. If you flip it three times and, say, obtained three heads, then the probability that the next flip is heads is just $1/2$. The forth flip does not depend on what happened previously.