 Solution

The kinship coefficient ${\varphi }_{XY}$ is the probability that one allele copy sampled uniformly from each of individuals $X,Y$, independently conditional on their copies, is identical by descent. The inbreeding coefficient $F_X$ is the probability that the two homologous allele copies within $X$ are identical by descent. For distinct offspring and older individual, Mendelian segregation givesFor the last identity, two independent selections choose the same copy with probability $1/2$, and choose distinct copies with probability $1/2$, in which case identity has probability $F_X$. An unrelated, noninbred founder has $F=0$ and self-kinship $1/2$, while distinct such founders have zero mutual kinship coefficient. The source's statement of zero founder kinship must be read as zero kinship between distinct founders; zero self-kinship would contradict the definition.