Jerry Lewis is throwing darts at the board pictured in Figure 1. It has the following properties:
- The board has radius \(r\);
- The board is partitioned into five rings, each with the same “thickness” \(r/5\);
- The board is mounted on a wall of area \(A\).
Jerry is not very skilled. His dart is guaranteed to reach the wall and stick someplace, but beyond that, the probability that it lands in any given region is just proportional to that region’s area.
Task: Derive a general formula for the conditional probability that a dart lands in ring \(i\) given that it hits the board and not the wall. Your formula should be a function of \(i=1,\,2,\,3,\,4,\,5\). Check that these probabilities are nonnegative and sum to one.