How Many Individuals Share an IBD Segment?

How often does one find IBD segments that are shared by more than two individuals? The probability of finding IBD segments increases with decreasing segment size (36). Thus, the smaller the IBD segments, the greater the chance of finding multiple individuals who share this segment. In a growing population with $N$ founding haplotypes, the probability that two randomly chosen haplotypes are IBD at a very short random segment (no recombination) is $1/N$ (17). The probability of $k$ individuals sharing a very small segment is $1/N^{k-1}$ . In a cohort of $M$ individuals, there exist $\genfrac{(}{)}{0pt}{1}{M}{k}$ sets of $k$ individuals resulting in an expected number of

$$\frac{1}{N^{k-1}} \ \genfrac{(}{)}{0pt}{0}{M}{k} \ = \ \frac{M}{k!} \ \frac{(M-1)}{N} \ldots \frac{(M-k+1)}{N}$$ (1)

segments that are shared by $k$ individuals. This number holds for one DNA location. Genetic drift may have led to more frequent segments than $1/N$ increasing the likelihood of finding IBD segments that are shared by multiple individuals. In summary, many IBD segments will be shared by more than two individuals if (a) the segments are small and (b) the cohort is large.