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The Odds of Being Related

Mathematically speaking, the odds are in your favor of being able to prove a relationship with just about any other person, living or deceased. In fact the biggest challenge may be tracing your lineage past your grandfather or great grandfather. It seems that the farther back one goes, the more likelihood you are of finding a pedigree that extends many generations. Your ancestors increase exponentially each time you move back another generation. It works likes this:

You have 2 ancestors, if you only go back one generation.
You have 6 total direct bloodline ancestors if you go back two generations.

The numbers aren't all that impressive, even at 10 generations (2,046 direct ancestors).
Even so, that's enough to be able to find a common Marcotte ancestor with TWO Canadian
Prime Ministers: Jean Chretien (direct descendant of Jacques Marcot) and
Louis Stephen St. Laurent (direct descendant of Nicolas Marcot).

At five generations more (15 generations) the number swells to 65,534 direct ancestors. Not bad. Surely, there's a celebrity in there somewhere.

At 20 generations back, you may be surprised to learn that you have a total of 2,097,150 total ancestors that are in your direct bloodline. Try the math yourself, if you don't belive me. Granted, it is somewhat of a challenge to trace one's line back 20 generations.

At some point, your direct ancestors are going to add up to a number that approximates or exceeds the total number of people that have ever lived (An estimate based on calculations by Harvard professor Nathan Keyfitz approximates 96 billion people). On more than one occasion, I have heard professional genealogists speaking on public radio or television say that at 25-30 generations (67,108,862 direct ancestors), any person alive should by pure mathematics have a common ancestor with any other known person, alive or dead.
To determine how many direct ancestors you have (known or unknown) at any given generation, just use the following formula:

Total direct ancestors = (2 to the nth power)- 2, where n=the number of generations (counting yourself).

50 generations of ancestors = 2 quadrillion+ ancestors
40 generations = 2 trillion+ ancestors
30 generations = 2 billion,147 million+ ancestors
20 generations = 2,097,150 ancestors,i.e. - [(2 to the 21st power) - 2] = [2,097,152)-2] = 2,097,150
15 generations = 65534 ancestors
14 generations = 32766 ancestors
13 generations = 16382 ancestors
12 generations = 8190 ancestors
11 generations = 4094 ancestors
10 generations = 2,046 ancestors
9 generations = 1022 ancestors
8 generations = 510 ancestors
7 generations = 254 ancestors
6 generations = 126 ancestors
5 generations = 62 ancestors
4 generations = 30 ancestors
3 generations of ancestors = 2 to the 4th power minus 2 = 14 ancestors. i.e. - [(2 to the 4th power) - 2] = [(16)-2] = 14
your 8 great grandparents, plus 6 from below
2 generations = 6 ancestors = your parents plus both sets of grandparents 2 to the
1 generations = 2 ancestors = your parents...i.e., [(2 to the 2 power) - 2] = [(4-2)-2] = 2

NOTE: There are, however, a couple of major flaws in the above calculation. First of all, most people have ancestors who pop up in more than one line of descent, i.e.- Ancestor Z's great great great granddaughter via son X marries Ancestor Z's great great great grandson via daughter W. If you have cousins who intermarried (and they tended to do that more often in the 17th and early 18th centuries), then your total number of ancestors would drop, sometimes significantly. The closer to your generation that it happened the greater the reduction of ancestors. The more often it happened, the greater still the reduction. If your great grandparents were second cousins, then at generation 6, you only have 124 direct ancestors, instead of 126, and at 7 generations back, 248 ancestors...
Generation 6 = only 2 less ancestors, so 124, instead of the 126 above.
Generation 7 = 248, i.e. - 6 fewer ancestors than above
Generation 8 = 496
Generation 9 = 992
Generation 15 = 63,488
Generation 20 = 2,031,616, still just 65,534 fewer ancestors
But, by Generation 30 = 2,080,374,784 ancestors: over 67 million fewer* than had the great grandparents not been 2nd cousins.

In my own lineage I have found at least 20 different ancestors who pop up in common in more than one line, mosly between 10 to 14 generations back. Another ancestor who pops up in two separate descents to me from 11 generations back removes another half million from the count of possible unique ancestors I could have had, and so on, and so on. Hmmm, only another 18 or so occurrences that I'd have to remove. I'm not willing to spend the time (I detest math) required to calculate how much that would reduce the total number of ancestors at generations 40 or 50, but I can easliy see how the total number could drop from a couple of quadrillion ancestors to about 70 trillion! Still, what's a quaddrillion or two plus or minus a few trillion.

The second flaw is that such total number of ancestors does not mesh with actual population statistics. The total population of the world did not exceed 1 billion people until the early 1800s. The period 1840-1850 is only 4 generations back in my family tree, a date range for which I have only direct 30 ancestors. In reality, the total number of a person's unique ancestors at 15 to 20 generations is much, much smaller due to the increasing frequency of common ancestors, as you move farther back in time. An analysis by Torben Andersen (of the Kellog School of Management at Northwestern University) of the recorded ancestry of John of Gaunt (1340-1425), for example, reflected 32,766 possible ancestors at 14 generations (15 generations counting John). However, only 13,923 of John's ancestors names were known to that generation, and of those 13,923, only 1,901 were unique (royal ancestries tend to include many more intermarriages than commoners, due to the desire to marry another royal, of which there are relatively few bloodlines from which to choose). Through statistical sampling, Torgen and other researchers/mathematicians estimate that most of us likely have only about 100,000 unique ancestors through 100 generations, and only about 25,000 at 30 generations. Conversely, the number of possible descendants from an individual at 30-35 generations back is an astronimical number, and high enough to ensure a common ancestor with virtually any other living individual. To illustrate this in a fairly dramatic fashion, view this chart of shared ancestral relationships between a couple of my own Marcotte ancestors (18 & 19 generations before me) whose descendants also include the current regents not just one or two, but of ALL the still-existing Western European monarchies. The genealogies of monarchs are always well documented, so they lend themselves well to such an example.

For an equally illustrative example, see my research with the group referred to as the Companions of William the Conqueror.

Bottom Line:...the next time some idiot cuts you off in traffic, or otherwise raises your blood pressure, be kind. Be sympathetic. Be depressed. He/She is probably your 26th cousin, 3-times-removed.

* The formula I used for the reduction for a common ancestral couple was using (2 to the g power) - 2, where g = the generation for which you are counting possible ancestors minus the number of generations that occurred since that particular common ancestry in 2 separate descents to you. Example: For 30 generations of ancestors, such a common ancestor pair who occurred twice in your line at 11 generations back in time from you removes 2 to the 19th power = 524,286 unique ancestors from your possible 30 generations pool. Until 10/16/2012, I had used an inaccurate formula for computing that number which exaggerated the variance caused by a conmon ancestor.

- Michael Marcotte

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