Poisson Distribution

p(20) = 0.0418, or about 1 chance in 24.

Explanation

We get this off the same Table as the last result. The problem is mere dictionary work.

Comment

There are no tricks here. But there are points of interest. It seems that the spread of possible results for this situation is considerable. To get a preliminary sense of how wide the meaningful part of the distribution is, we may add the probabilities for all outcomes between p(10) and p(20), a range which has the rate 15 in the middle. We get the total probability p(10 ~ 20) = 0.8470, or about 85% of all possible results. That was a rough result. We now try to refine it a little.

What, we next ask, is the range which most efficiently includes 99% of the possible outcomes? A first approximation is to eliminate all values for which the table gives probabilities of 0.0000; this tells us that the detectable values lie within p(3) and p(33) inclusive; note that 3 and 33 are not equidistant from 15, reflecting the skew nature of the Poisson distribution. But even those visible values are highly unlikely (0.0002 and 0.0001, respectively). If from both ends of that range, we eliminate values of roughly the same size until the cumulative size of the eliminated values is just below 1%, we will have the answer to our question. It turns out that the range we want is from p(6) to p(25) inclusive. The excluded values total 0.0089, which is less than 1%. It is therefore 99% certain, which means operationally certain, that between 6 and 25 males aged 42, and insured by Malingerer's Life, will die within the next year.

Within this most efficient range there are 9 values below the base value 15, but 10 values above it, reflecting the Poisson skew toward the non-zero side of its thematic value r. We will meet this fact again; in fact, we will meet it every time we consider the full detail of any Poisson frequency profile.