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jmc

counting and probability intermediate

Problem

Let be the set of all sides and diagonals of a regular pentagon. A pair of elements of are selected at random without replacement. What is the probability that the two chosen segments have the same length?
Solution
In a regular pentagon, there are sides of the same length and diagonals of the same length. Picking an element at random will leave 4 elements with the same length as the element picked, with total elements remaining. Therefore, the probability that the second element has the same length as the first is simply
Final answer
\tfrac{4}{9}