How Hutu's a Tutsi?

There has been violence in eastern DR Congo between Hutu ethnic groups and Tutsi groups in the last decade, transferred from previous conflicts in Rwanda. Burundi has suffered similar conflicts along the same lines. Since the ethnic identification is so important in these conflicts, it is worth considering how meaningful it is by looking at whether a Hutu has substantially different ancestors from a Tutsi.

One legend or myth of the Tutsis' migration is that they arrived in Central Africa after the Hutu, coming from the Horn of Africa around 1600. I do not know if this is correct or not; for the purposes here its relevance is only that it gives two initial disjoint groups: Hutus numbering p people and Tutsis numbering q people. If the initial groups are not as clear cut as the origin story implies, then the mixture today of ethnic groups is even greater than the following results already imply.

Suppose that a person in either ethnic group reproduces in their own group with a high probability, say 0.8 at every generation, and being a Hutu or Tutsi means having only ancestors from this group. Then the probability of someone today with Rwandan or Burundian ancestry having exclusively Hutu or Tutsi ancestors is the probability that their father reproduced with someone from the same ethnic group times the probability that their paternal grandfather did the same times the probability that their paternal great grandfather did the same and so on back to 1600, or 16 generations assuming reproduction at age 25. The probability is 0.8^(16-1) = 3.5 percent, which is pretty small.

The 0.8 probability is unrealistically high because as the number of people who are ethnically of a single heritage declines over time, the chances of someone from this group meeting someone else from this group declines. Moreover, the percentage implies a deliberate process of reproductive selection from the exclusive group, which will be subject to increasing error over time due to uncertainties over heritage and the unreliability of identification on physical characteristics (a former leader of Burundi established his Tutsiness by virtue of the shape of his father's nose, if I remember one account correctly, with his own features being suspiciously Hutu). Finally, people often prefer partners with some physical differences from themselves, presumably out of their ethnic group.

So suppose that, at least after the initial few generations, partners are selected at random from the population. Then after n generations, the number of ancestors from the initial groups is 2^(n-1) (2 parents, 2*2 grandparents, 2*2*2 great grandparents, and so on). The probability of having r Hutu ancestors is from the binomial distribution, equalling C(2^(n-1), r)*(p/(p+q))^r*(q/(p+q))^(2^(n-1)-r). The probability of no Hutu ancestors is (q/(p+q))^(2^(n-1)). Taking (q/(p+q)) as 0.1, and n as 15, it equals 0.00000000000000000000000000000000...000000001 percent where the ... denotes 30000 zeroes. The chance of having no Tutsi ancestors is similarly miniscule.

From the properties of the binomial distribution, the expected number of Hutu ancestors is (p/(p+q))*2^(n-1). The standard deviation of the distribution is the square root of (p/(p+q))*(q/(p+q))*2^(n-1). The binomial distribution can be approximated by the normal distribution to calculate likely bounds for the number of Hutu ancestors. It is in the region of [(p/(p+q))*2^(n-1)-3*square root of (p/(p+q))*(q/(p+q))*2^(n-1), (p/(p+q))*2^(n-1)+3*square root of (p/(p+q))*(q/(p+q))*2^(n-1)] with 99 percent probability. As the square root of 2^(n-1) grows more slowly than 2^(n-1) itself, the ratio [Hutu ancestors of one person today / Hutu ancestors of another person today] is close to one. In other words, the leader of the Hutu militias in eastern DR Congo has a similar number of Hutu and Tutsi ancestors to the leader of the Tutsi armies.

So what explains the ethnic identification? It may be socially invented in recent years, or it may be the result of arbitrary allocation in response to physical characteristics in the last few family generations.

I think these sums are OK. The huge difference between 0.8 and the 2^(n-1) calculations arises because the 0.8 selection implies that people are willing to work very hard to select within their own ethnic group, and accept possible inbreeding.