Talk: Probability

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I've moved the existing talk page to Talk:Probability/Archive1, so the edit history is now with the archive page. I've copied back the most recent thread. Hope this helps, Wile E. Heresiarch 04:44, 10 Aug 2004 (UTC)

Law of Large Numbers

First, good job on the entry to all....

Now, the Law of Large Numbers; though many texts butcher this deliberately to avoid tedious explanations to the average freshman, the law of large numbers is not the limit stated in this entry. Upon reflection, one can even see that the limit stated isn't well-defined. Fortunately, you present Stoppard's scene that so poignantly illustrates the issues involved with using the law of large numbers to interpret probabilities; the exact issue of it not being a guarantee of convergence. I have an edit which I present for discussion:

... As N gets larger and larger, we expect that in our example the ratio N_H/N will get closer and closer to the probability of a single coin flip being heads. Most casual observers are even willing to define the probability Pr(H) of flipping heads as the mathematical limit, as N approaches infinity, of this sequence of ratios:

<math>\Pr(H) = \lim_{N \to \infty}{N_H \over N} <math>

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). Furthermore, mathematically speaking, the above limit is not well-defined; the law of large numbers is a little more convoluted and dependent upon already having some definition of probability. The theorem states that, given Pr(H) and any arbitrarily small probability ε and difference δ, there exists some number n such that for all N > n,

<math>Pr\left( \left|\Pr(H) - {N_H \over N}\right|>\delta \right) <\epsilon<math>

In other words, by saying that "the probability of heads is 1/2", this law asserts that if we flip our coin often enough, it becomes more and more likely that the number of heads over the number of total flips will become arbitrarily close to 1/2. Unfortunately, this theorem says that the ratio will probably get close to the stated probability, and provides no guarantees of convergence.

This aspect of the law of large numbers is sometimes troubling when applied to real world situations. ...

--Tlee 03:44, 13 Apr 2004 (UTC)

Since N_H/N is just the sample mean of a Bernoulli random variable, the strong law of large numbers should guarantee the convergence of N_H/N to the mean, Pr(H). That is, convergence will occur almost surely, or equivalently

<math>\Pr\left( \lim_{N\rightarrow\infty} {N_H \over N} = \Pr(H) \right) = 1.<math>

Nonetheless, I agree that the way probability is `defined' in the current version of the article needs some refinement, and other than the above comment, I like what you've got, Tlee. I'd say just go ahead and make the edit!

--Ben Cairns 23:47, 15 Apr 2004 (UTC)

Poker

This article talk about how probability applies to poker.