Law of truly large numbers

[skyblue1]

The law of truly large numbers, attributed to Persi Diaconis and Frederick Mosteller, states that with a sample size large enough, any outrageous thing is likely to happen.

Because we never find it notable when likely events occur, we highlight unlikely events and notice them more. The law seeks to debunk one element of supposed supernatural phenomenology.

For a simplified example of the law, assume that a given event happens with a probability of 0.1% in one trial. Then the probability that this unlikely event does not happen in a single trial is 99.9% = 0.999.

In a sample of 1000 independent trials, the probability that the event does not happen in any of them is , or 36.8%. The probability that the event happens at least once in 1000 trials is then 1 − 0.368 = 0.632 or 63.2%. The probability that it happens at least once in 10,000 trials is .

This means that this "unlikely event" has a probability of 63.2% of happening if 1000 chances are given, or over 99.9% for 10,000 chances. In other words, a highly unlikely event, given enough tries, is even more likely to occur.

The law comes up in criticism of pseudoscience and is sometimes called the Jeane Dixon effect (see also Postdiction). It holds that the more predictions a psychic makes, the better the odds that one of them will "hit". Thus, if one comes true, the psychic expects us to forget the vast majority that did not happen.

Humans can be susceptible to this fallacy. A similar manifestation can be found in gambling, where gamblers tend to remember their wins and forget their losses and thus hold an inflated view of their real winnings.

Steven Novella describes this as the "lottery fallacy":

It is also the lottery fallacy. If we hold a world-wide lottery and only one human in the 6.5 billion wins, the odds of that person winning is very small. But someone had to win. Chopra and Lanza are arguing that the winner could not have [won] by chance alone, because the odds were against it.


http://en.wikipedia.org/wiki/Law_of_truly_large_numbers

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[skyblue1]

A set of mathematical laws that are called the Improbability Principle tells us that we should not be surprised by coincidences. In fact, we should expect coincidences to happen. One of the key strands of the principle is the law of truly large numbers. This law says that given enough opportunities, we should expect a specified event to happen, no matter how unlikely it may be at each opportunity. Sometimes, though, when there are really many opportunities, it can look as if there are only relatively few. This misperception leads us to grossly underestimate the probability of an event: we think something is incredibly unlikely, when it's actually very likely, perhaps almost certain.

How can a huge number of opportunities occur without people realizing they are there? The law of combinations, a related strand of the Improbability Principle, points the way. It says: the number of combinations of interacting elements increases exponentially with the number of elements. The “birthday problem” is a well-known example.

The birthday problem poses the following question: How many people must be in a room to make it more likely than not that two of them share the same birthday?

The answer is just 23. If there are 23 or more people in the room, then it's more likely than not that two will have the same birthday.

Now, if you haven't encountered the birthday problem before, this might strike you as surprising. Twenty-three might sound far too small a number. Perhaps you reasoned as follows: There's only a one-in-365 chance that any particular other person will have the same birthday as me. So there's a 364/365 chance that any particular person will have a different birthday from me. If there are n people in the room, with each of the other n − 1 having a probability of 364/365 of having a different birthday from me, then the probability that all n − 1 have a different birthday from me is 364/365 × 364/365 × 364/365 × 364/365 … × 364/365, with 364/365 multiplied together n − 1 times. If n is 23, this is 0.94.

Because that's the probability that none of them share my birthday, the probability that at least one of them has the same birthday as me is just 1 − 0.94. (This follows by reasoning that either someone has the same birthday as me or that no one has the same birthday as me, so the probabilities of these two events must add up to 1.) Now, 1 − 0.94 = 0.06. That's very small.

Yet this is the wrong calculation to consider because that probability—the probability that someone has the same birthday as you—is not what the question asked. It asked about the probability that any two people in the same room have the same birthday as each other. This includes the probability that one of the others has the same birthday as you, which is what I calculated above, but it also includes the probability that two or more of the other people share the same birthday, different from yours.

This is where the combinations kick in. Whereas there are only n − 1 people who might share the same birthday as you, there are a total of n × (n − 1)/2 pairs of people in the room. This number of pairs grows rapidly as n gets larger. When n equals 23, it's 253, which is more than 10 times as large as n − 1 = 22. That is, if there are 23 people in the room, there are 253 possible pairs of people but only 22 pairs that include you.

So let's look at the probability that none of the 23 people in the room share the same birthday. For two people, the probability that the second person doesn't have the same birthday as the first is 364/365. Then the probability that those two are different and that a third doesn't share the same birthday as either of them is 364/365 × 363/365. Likewise, the probability that those three have different birthdays and that the fourth does not share the same birthday as any of those first three is 364/365 × 363/365 × 362/365. Continuing like this, the probability that none of the 23 people share the same birthday is 364/365 × 363/365 × 362/365 × 361/365 … × 343/365.

This equals 0.49. Because the probability that none of the 23 people share the same birthday is 0.49, the probability that some of them share the same birthday is just 1 − 0.49, or 0.51, which is greater than half.

http://www.scientificamerican.com/articl...e-lottery/

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[Gopher Gary]

Because we never find it notable when likely events occur, we highlight unlikely events and notice them more.

People highlight and notice unlikely events more because they're interesting, and they're interesting because they're not predictable. Not long ago I expressed, I don't know why some people are interesting and most others aren't. I think that's it.