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#3
12-04-2005, 06:35 AM
 Siegmund Senior Member Join Date: Feb 2005 Posts: 415
Re: Poisson and Approximating Binomial

Yes, the Poisson distribution can be used to make such approximations. The rarer the event the better the approximation.

It is "simpler" in the sense that, until the last ten years or so, it was the only practical way of doing these at all, and it's still very handy for doing ballpark estimates in your head. (And it's handy for avoiding underflow errors, if you're working on a machine with limited precision.)

Let's look at two recent threads in this forum. Scroll down to "No Royal in 100k hands?" We're told that chance of going 100,000 hands (if we take every hand to the river )without holding a Royal is 1 in 30940.

How do you calculate (30939/30940)^100000 by hand? You take logarithms. ln P = 100000 ln 30939/30940 = 100000 ln (1-1/30940) = 100000 * (- 1/30940 + 1/30940^2 etc), you approximate by dropping the higher-order terms from the series expansion of the log, and you have ln P ~ -100000/30940, or P=e^(-100000/39040).

The Poisson formula just generalizes this, to P(something never happens) = e^-(#times something is expected to happen), and saves you from similar but harder series.

In the Royal Flush thread, e^-3.23206 = 0.03948, and we expect the approximation to be good to 4 decimal places since the neglected term in the series was 1/30940 as big as the included term.

By comparison, in the thread "what are the odds of 100 hands with no PP?" we are asked to calculate (16/17)^100. Here the Poisson approximation is e^-100/17 = e^-5.8824 = 0.279%, while the actual answer was 0.233%. This is not surprising since the neglected term is now only 1/17 as small as the term we considered, and an error appears in the second significant digit.

The number of digits you can trust in your approximation is, then, controlled by how unlikely the event is to occur on a single trial. In your case, you can expect using the Poisson approximation instead of an exact calculation to give you 5 good decimal places.

3) Yes; the variance and mean of a Poisson distribution are equal. If you run many series 30M trials with p=10^-5, your estimates are going to be centered on 300 with a standard deviation between 17 and 18.

This is the other reason why the Poisson approximation is used. Since the experiment itself is going to only provide you an answer with two significant digits, you needn't lose any sleep at all about the errors in fifth digit from the approximation.