NOTE: The original version of this post used an inappropriate simplification of the math. Ann Turner, Louis Kessler, and Andrew Millard were all kind enough to point out my error, which I have incorporated into this revised version. Many thanks to the three of them for helping to make this a much better post!
I’m in a math kind of mood today, so let’s talk about MPEs (mis-attributed parentage events, usually of the father) and probability. First, a quick overview of the two cardinal rules of probability: the AND rule and the OR rule.
Math Is Fun!
The AND rule goes like this: the probability that two independent events will both happen is the probability of the first times the probability of the second. If I flip a coin that has a 50-50 chance of being heads or tails, the probability that I will get heads two times in a row is 0.5 x 0.5 = 0.25. (Go ahead, try it: flip a coin twice in a row, repeat 100 times, and tally up how many times you get heads–heads; it should be pretty darned close to 25%.)
The OR rule is: the probability that either of two independent events will happen is the probability of the first plus the probability of the second. The chance that I will flip either heads or tails when I flip my coin is 0.5 + 0.5 = 1.0. In other words, there’s a 100% chance that I’ll flip one or the other.
I know what you’re thinking: Does this have anything to do with DNA, or does she just like to hear herself type? (You would add those probabilities, by the way.) The answer is: Both! (Multiply ’em.)
MPEs in Your Family Tree
Has a genealogist ever told you that they don’t need to take a DNA test because they’ve got a solid paper trail back to the 1600s. They’re wrong, and here’s why: even in families where there is no reason to suspect misattributed paternity, its rate of occurrence is about 1–2%. (The ISOGG Wiki has a good overview of studies on misattributed paternity rates.) That rate is not universal—it’s affected by parental age, marital status, and socioeconomics, and it’s higher in some countries than in others—and it doesn’t include other forms of misattributed parentage, like an undocumented adoption, a step-parent who is mistakenly assumed to be the biological parent, or grandparents who raise their grandchild as their own.
Consider a “family tree” with just me in it. Without any DNA evidence proving otherwise, there is a 1–2% chance that one of my parents is misidentified. For simplicity, let’s use 2%. That means there’s a 98% chance, or 0.98 probability, that both of my parents are who I think they are.
Now, let’s add my parents to the mix. This looks like an OR situation: the MPE could be with me or my father or my mother. If we add the numbers, we get 2% + 2% + 2% = 6% chance, or 0.06 that one of us has misattributed parentage. Alternately, we could ask “What is the chance that there is no MPE among these three people?” That would be an AND situation, because the answer requires no MPE for me and no MPE for my father and no MPE for my mother. The math is 0.98 x 0.98 x 0.98 = 0.983 = 0.941192, or 94.1% chance.
Wait, what? Those two different approaches give different answers! How can that be? Turns out I made an error using the OR strategy, because the three different events are not exclusive of one another. That is, more than one of us could have misattributed parentage. The AND strategy is the better approach here. (Thank you to Ann Turner, Louis Kessler, and Andrew Millard for pointing out my error.)
There are seven people in the tree once I include my four grandparents, so the overall chance that none of us have an MPEs is 0.987 = 0.868, or 86.8%.
Extending back to my eight great grandparents, the math is 0.9815 = 0.739, or 73.9%.
By the time an average tree includes eight generations, there’s almost no chance that every single parent is correctly identified. So tell me again why you don’t need to check your paper trail with DNA. Just as you can’t prove biological relationships with DNA alone, you can’t do it solely with documents. We need both. (Blaine Bettinger makes this point with an excellent example comparing a niece to the identical twin of her mother in the Facebook group Genetic Genealogy Tips & Techniques. You can read the thread here.)
My Tree and MPEs
I have done autosomal DNA tests on myself and my parents at AncestryDNA (mom) and 23andMe (dad), and they both match me as parent–child. Because they match me across my entire genome and neither of them was an identical twin, I know that they are my parents and that there is a 0% chance that I’m the product of an MPE.
My parents match with the expected amounts of DNA to first cousins and/or first cousins once removed through all four of their grandparents, so I can consider my relationships to my great grandparents, and by extension my grandparents, supported as well. Similarly, DNA matches to 2nd, 3rd, and more distant cousins indicate that all four of my great grandparents were almost certainly who I thought they were.
With autosomal DNA evidence, my family tree going back three generations has a roughly 0% chance of including an MPE. In addition, a yDNA test at Family Tree DNA on my father confirmed that our surname lineage is Larkin.
By virtue of DNA testing, I’ve lowered the chances of an MPE in this part of my tree from 26.1% (100% – 73.9%) to nearly zero. (I use the ≈ symbol in the figure to indicate that it’s “about 0%”, because some alternate scenarios could have led to the DNA matching patterns I see. For example, one of my ancestresses could theoretically have had a child with her husband’s brother, and the DNA might not be able to tell.)
In fact, the great grandfather who is marked with a yellow star in the tree was sort of an MPE. “Sort of” because documentation listed the names of his parents but, prior to having DNA evidence, my family believed that he’d been born in France and came to America as a runaway. Using DNA testing, I was able to show that his parents were locals but not married to one another. That information, in turn, allowed me to extend that branch of my family tree back several more generations. You can read about that search here.
Yes, Even If You Think You Don’t
No amount of paper can prove that the father listed on a record is the biological parent, birth certificates are often amended or outright forged in adoption cases, and one can falsely infer who a child’s parent was from census records when the other parent has remarried. For all of these reasons, DNA is needed to support the conclusions we draw from documentary evidence.
One Last Comment
As a final aside, the 2% MPE rate in this post is just an approximation used to demonstrate the math. It’s not meant to be an accurate representation of the expected MPE rate in my family or any other. Some families will have higher rates, others lower. And the truth is, we don’t know what the true MPE rate is overall. In fact, this is an area where good genealogical skills can contribute to academic research on the subject.