Note: In this analysis, the symbol ^ means to the power of, ie x^y denotes x to the power of y.
Let us identify a sample comprising a whole generation of your ancestors living n generations ago, (ie 2^n individuals),and let us identify a particular individual standing among them as person X.
We already know, that if the sample 2^n is big enough, then X will appear among them more than once,ie you are descended from X two or more different ways.
The chance of any one arbitrary person in the 2^n sample being X is 1/p, where p is the entire national population (as he could just as easily be anybody else).
Therefore,the chance that this arbitrary person in the 2^n sample is not X, is 1-(1/p)
Therefore,the chance that none of the people in the 2^n sample is another X is
(1-(1/p))^(2^n)
Therefore, the chance that this in turn is not so,and that somebody in the 2^n sample is another X,is
1-((1-(1/p))^(2^n))
Therefore, the number of people in the 2^n sample which are X,in other words the number of times X appears in the sample,is
(1-((1-(1/p))^(2^n)))x(2^n)
Subtracting this quantity from the 2^n sample, the number of different ancestors in the 2^n sample is therefore,
(2^n)-(2^n)x(1-((1-(1/p))^(2^n)))
=(2^n)x(1-(1-((1-(1/p))^(2^n)))
=(2^n)x(1-1+(1-(1/p))^(2^n))
=(2^n)x((1-(1/p))^(2^n))
Note: There are a number of occasions in this derivation where the quantity 1 has to be added to or subtracted from a much larger quantity. I have omitted this in order to avoid unneccessary complication. The error involved is negligible.
Measurements of many family trees has shown that the average length of one generation is 33 years.Also,on our time scale we shall count back from 1960,the average birth year for all people presently living.
n Number of different ancestors % of population
1 2 -
2 4 -
3 8 -
4 16 -
5 32 -
6 64 -
7 128 -
8 256 -
9 512 0.01 %
10 1024 0.02 %
11 2047 0.05 %
12 4092 0.1 %
13 8176 0.2 %
14 16318 0.4 %
15 32412 1.08 %
16 64120 2.14 %
17 125,468 4.2 %
18 229,934 11.5 %
19 403,362 20.2 %
20 620,655 31.0 %
21 734,733 36.7 %