# What is the probability of an offspring sharing identical HLA typing as one of their parents?

DISCLAIMER: I have yet to thoroughly study HLA 100% to the bone, and hence I won't know everything about it at the back of the hand.

Recently I came across this information on a San Francisco hospital website claiming the chance of a full HLA match between 2 randomly selected people is 1/1,000,000 and the chance of a full HLA match between a child and one of their parents is 1/200.$$^{}$$

What struck me as odd is the 1/200 chance of a parent matching their offspring's HLA, which seemed too high of an estimate. The immediate intuition is for a child to have identical HLA typing as one of their parents, their parents need to share some HLA haplotypes, which on its own should be a rare occurrence. Hence I decided to attempt verification of this number to confirm if it was within the realm of possibility

According to Stanford$$^{}$$, the HLA cell surface receptors that mix during inheritance are HLA-A, HLA-B in MHC I, and HLA-DR in MHC II. A set of these 3 forms a haplotype. [HLA-C and HLA-DQ are probably involved too, but we'll omit them for simplicity's sake.]

For our estimation, we use data from the same Stanford source$$^{}$$ and find there are 59 HLA-A serotypes, 118 HLA-B serotypes, and 124 HLA-DR serotypes. This gives us $$59\times118\times124=863288$$ different possible haplotypes.

To get the probability of 2 parents sharing at least 1 common haplotype, we do some math: $$\frac{1}{863288} \times \frac{863287}{863288} + \frac{1}{863288} \times \frac{863287}{863288} + 2 \times \frac{1}{863288} \times \frac{1}{863288} = 1/431644$$

The above is assuming the parents have met randomly in the world and no inbreeding has taken place. Inbreeding will pretty much ensure 2 parents share common HLA haplotypes.

This means the chance of an offspring to even possibly have the same HLA typing as one of their parents is 1 in 431,644, which is already way lower than the 1 in 200 chance stated on the hospital's website.

Just for calculation sake, the precise chance of an offspring sharing the same HLA typing with either one of their parents after accounting for different possible haplotype combinations produced by the parents would be: $$100\% \times \frac{1}{863288}^2 \times 2 + 25\% \times \frac{1}{863288} \times \frac{863286}{863288} \times 2 + 50\% \times 2 \times \frac{1}{863288}^2 = 5.792 \times 10^{-7}$$

An exceedingly small chance.

Issues with my estimation:

There were 2 assumptions made during this estimatey computation. I didn't take into account the weighted and uneven distribution of the haplotypes, and just assumed they are all statistically evenly distributed. Some haplotypes are more common than others, and this factor was not considered during the estimation. If existing data on haplotype occurrence is used, it will likely affect the estimation quite a bit, however, this is highly difficult as the data varies by region and ancestry.

I also did not take Genetic matchmaking into account, which is a factor that will lower the chance of 2 parents sharing common HLA haplotypes.

If exact computation for the probability proves to be difficult, what would be a reasonable estimation on the order of magnitude of said probability? Is 1/200 within the realm of possibility once you take into account the uneven distribution of haplotypes caused by certain more common ones?

Sources:

$$^{}$$https://www.ucsfbenioffchildrens.org/education/types_of_bmt_donors/#2

$$^{}$$http://web.stanford.edu/dept/HPST/transplant/html/hla.html