Consider a bifurcating tree with $n>1$ leaves (or "individuals") produced by a branching process. All individuals derive from a single founder individual at the root of the tree, such that we can trace back the "ancestry" of each current individual to this single common ancestor.
Any two individuals are connected through a most recent common ancestor at which their lineages join in a node in the tree, whose lineage joins with another lineage and so on, up the the founder individual.
The height of the tree is represented by "time" as a continuous variable. The time during which there are $j$ distinct ancestors that have not yet joined in the common ancestor of the tree is exponentially distributed with parameter ${j(j-1)/2}$, where ${2 \leq j \leq n}$.
What is the probability that the most recent common ancestor of a randomly selected pair of individuals is also the common ancestor of the tree?