In tennis, a server should win a larger percentage of second-serve points compared to first-serve points; that much we know.  Partly that’s because a server optimally serves more faults (serves that land out) on first serve than second serve.  But what if we condition on the event that the first serve goes in? Here’s a flawed logic that takes a bit of thinking to see through:

Even conditional on a first serve going in, the probability that the server wins the point must be no larger than the total win probability for second serves. Because suppose it were larger.  Then the server wins with a higher probability when his first serve goes in.  So he should ease off just a bit on his first serve so that a larger percentage lands in, raising the total probability that he wins the point.  Even though the slightly slower first serve wins with a slightly reduced probability (conditional on going in) he still has a net gain as long as he eases off just slightly so that it is still larger than the second serve percentage. Indeed the lower probability of a fault could even raise the total probability that he wins on the first serve.

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