Why fair choices produce uneven matchboxes
Imagine placing one match in a box after every coin-flip-like choice: left or right, each with a one-in-two chance. The choices are fair, but fairness applies to each turn—not to the totals after a run of turns. Nothing forces a later choice to “make up” for an earlier streak.
After several left choices in a row, the left box has a lead. Alternating choices would keep the piles close, but a random sequence does not deliberately alternate. Its running difference behaves like a random walk: each turn nudges the count toward one box or the other, and chance can temporarily carry it far from even.
What the negative binomial curve counts
A negative binomial distribution answers a particular question about that sequence: if we keep making fair choices until one box has been chosen a fixed number of times, how many choices of the other box will have occurred along the way? Those extra choices are the “leftovers” in the count. Different possible totals have different probabilities, which form the curve.
This is a different lens from asking how uneven the boxes are after a fixed number of turns. That question is described by a binomial distribution. The negative binomial view instead fixes a target count for one side and lets the total number of turns vary.
The useful surprise
The curve does not say that a long streak is impossible, or that the boxes must eventually balance. It assigns likelihoods to the different ways a fair sequence can unfold. Even a perfectly even chance at every turn can leave one box with a noticeable surplus—because random fairness is a rule for each choice, not a promise about the final piles.
