p(4T~10T) = 0.1209, as against p = 0.1428 for the previous Poisson calculation. The Poisson figure happens to be about 18% too high.Explanation
For an exact Binomial calculation (for those who skipped the Binomial chapter), for each probability we must multiply together 0.8 (for H) and 0.2 (for T) as many times as those results appear in a given outcome. Then we multiply that probability by the number of ways of getting that outcome. These are the Binomial Coefficients, and they can be calculated as a Combination problem, or simply read off the Arithmetic Triangle (the 1, 10, etc row is the one you want). That computation is shown below, without rounding, together with the corresponding figures according to the Poisson approximation (given to four places).
Binomial Poisson Coeff Frequency Product Value 1 p(0T) 0.1073741824 0.1073741824 p(0T) 0.1353 10 p(1T) 0.0268435456 0.268435456 p(1T) 0.2707 45 p(2T) 0.0067108864 0.301989888 p(2T) 0.2707 120 p(3T) 0.0016777216 0.201326592 p(3T) 0.1804 210 p(4T) 0.0004194304 0.088080384 p(4T) 0.0902 252 p(5T) 0.0001048576 0.0264241152 p(5T) 0.0361 210 p(6T) 0.0000262144 0.005505024 p(6T) 0.0120 120 p(7T) 0.0000065536 0.000786432 p(7T) 0.0034 45 p(8T) 0.0000016384 0.000073728 p(8T) 0.0009 10 p(9T) 0.0000004096 0.000004096 p(9T) 0.0002 1 p(10T) 0.0000001024 0.0000001024 p(10T) --
The respective sums for p(4T) through p(10T) are Binomial 0.1208738816 versus Poisson 0.1428, and these, both rounded to 4 decimals, are accordingly given in the answer space, above.
For guidelines about when to use Poisson to approximate Binomial, see the Lesson. As an addition to those guidelines, we note from the above table that the sum of several Poisson probabilities, which is what we have here, is a better match for the Binomial reality than are many of the single Poisson probabilities (some of which are off by a factor of 10 or more). The fact that the Poisson approximation is sometimes higher and sometimes lower than the Binomial figure means that the sums tend to average out.
There should be no doubt that the Binomial side of the table above is the real side. This is a classical Binomial problem, and it does not really fit the Poisson pattern. It is merely that we can sometimes, and for certain rough purposes, get away with behaving as though it did. The present purpose is sufficiently rough. We may be sure that Jimmie is going to lose. The question of exactly by how much he loses can fairly be called academic. When academic accuracy is desired, go Binomial.
Statistics is Copyright © 2001- by E Bruce Brooks
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