A rare state has a small weight relative to the total, so its probability and expected count are both small fractions of the population. With weights 1 and 7 the rare-state probability is 1/8, and population 40 gives expected rare count 40 times 1/8 = 5 exactly for the stated toy ensemble ledger; real rare-state estimates need enough samples, bias checks, missed-event handling, and measurement uncertainty.
A rare state is represented by a smaller finite weight. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Rare means fewer counted states
Weights 1 and 7 make probability 1/8 for the rare state.
1/8=81
The rare expected count stays exact
In a population of 40, the rare expected count is 5.