Changing the state fraction against a fixed population shows expected count as a checked product. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
One state out of eight gives five particles
The moving state has weight 1 out of total weight 8. In a population of 40, the expected moving count is 5.
40⋅81=5
Two states out of eight gives ten particles
Only the moving weight changes to 2. The probability is 1/4, so the expected moving count becomes 10.
40⋅41=10
Three states out of eight gives fifteen particles
With moving weight 3 out of 8, the expected count rises to 15. Three rows make fraction times population visible.