Poisson Distribution

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.

For an exact Binomial calculation (for those who skipped the Binomial chapter, or those for whom, as of 2013, the Binomial chapter is no longer online), 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, here salvaged from the Binomial chapter j(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 Poisson 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.