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

Question

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. . . .

"But squares that have already had several hits, I mean!"

"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."

Answer 1

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.

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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.

Answer 2

In a Poisson process, events occur randomly in time (or in a certain range) at a certain average rate and probabilities of the number of events that occur over a given range have a Poisson distribution. There is also, and most importantly here, an independence assumption that events occurring in one time period no not influence events that occur in another, disjoint time period.

With regard to a bomb hitting a particular location, even if that location has been hit many times already, the chance that it is hit in some future time is the same as if that location had never been hit (or hit only once, etc...). It does not matter what went on before, the probability of a bomb hitting a particular location within a particular time frame is always the same. The bombs have no "memory" of where others have hit, there is no "link" between any two bomb events, and conditioning on what happened in the past does not give you any information as to what happens in the future ("conditioning" means to assume something happened in the past, and calculate a probability based on that assumption).

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.

The Monte Carlo Fallacy is, as mentioned by Chris, is also named the Gambler's Fallacy. The fallacy is of the form "the first three flips of the coin were all heads, so the forth flip should be tails", because tails is "due" somehow (or that the forth flip will be heads, because it seems the coin is biased towards heads based on what happened previously).