A Non Paternal Events or "NPE" occurs when the ancestry of a person’s Y chromosome and surname go in different directions. A person named “Smith,” for instance, might have a Y chromosome that is clearly “Jones.” A in the Y chromosome might have occurred 1,000 years ago, 100 years ago, or with the one's own birth.

Such non paternal event can occur when, for eaxmple:-

- an adopted male takes the surname of his adoptive family.
- a male child takes his step-father’s surname
- a male child takes his mother’s surname
- on "extramarital event" (infidelity)
- Man changed name - various reasons

There's no agreement about the rate at which these occur. Read the literature avalable and you can see figures between 2% and 10% quoted for any one generation.

Adoptions were quite common in every age: parents died by disease or war and a relative took in the children and raised them with their name, daughters had children out of wedlock and the grandparents (or other relatives) raised the children as their own. Very young mothers of first-child sons in the line could be indicators for a higher probability of this phenomenon.

A paper by Anderson, Kermyt G. 2006. *How well does paternity confidence
match actual paternity? Evidence from worldwide nonpaternity rates. Current
Anthropology 48(3): 511-518. bar*. concludes that the median nonpaternity
rate is 3.3%.

Mathamatically the probability distribution of an NPE within a given number
of generations, n, is a geometric distribution with success parameter (probability
of an NPE), p, where the distribution function is given by

P(n) = p + p*(1-p) + p*(1-p)^2 + ... + p*(1-p)^(n-1)

My interest was to see the probability of a NPE in the last 10 generations.
In other words since about 1650 as one can expect 30 years between generations.

For p = 0.01 and n = 10

P = 0.0857 or about 9%

For p = 0.03 and n = 10

P = 0.2624 or about 26%

For p = 0.05 and n=10

P = 0.4026 or about 40%

For p = 0.07 and n=10

P = 0.0516 or about 51%