Probability of Third Dart Being Worse Than First

This question considers three independent attempts at hitting the center of a dartboard and asks about the relative ranking of their distances from the center, given partial information about the first two throws. The setup is purely probabilistic: no geometry of the board or skill model is needed beyond identical, independent performance and no ties. It is a clean example of reasoning about random rankings under a conditioning event, and would be typical in interviews that focus on probability intuition and exchangeability rather than heavy computation.

Solving it leans on symmetry of order statistics for i.i.d. samples, conditional probability, and careful counting of consistent orderings. The key technique is to recognize that only the relative ranks of the three distances matter, not their actual values or underlying distribution, as long as the throws are identically distributed and continuous. An interviewer is looking for a candidate to formalize this symmetry argument, avoid unnecessary modeling, and correctly enumerate the possible rankings after conditioning. Clear explanation, correct conditioning, and recognition that labels are interchangeable are more important than algebraic manipulation.

What it tests

When dealing with problems involving the random ordering or ranking of objects (such as distances, times, or scores), the key structure is the symmetry of all possible orderings under the assumption of identical, independent distributions. If no information distinguishes the objects except for imposed conditions, each ordering is equally likely. Conditioning on a partial ranking (such as one item being better than another) restricts the set of possible orderings, but within the remaining set, all are still equally likely. This symmetry allows us to count favorable outcomes among the valid ones without worrying about the underlying distributions, as long as ties are impossible and the process is fair. The principle holds because the process is exchangeable: relabeling the objects does not change the probability structure, so all orderings are a priori equally probable.