Solution to: Miss the Mark
Each shot has a chance of 1/3 to be the best. This is because the result of a shot is not dependent on an earlier shot. The chance that the third shot is not the best shot is 1 minus the chance that the third shot is the best shot. Therefore, the answer is 2/3.
You can also look at the possible outcomes. Call the three successive arrows A, B and C. We know that arrow B is further from the bull's-eye than arrow A, so after shooting arrow C the following three outcomes are possible:
For two of the three outcomes, arrow C is worse than arrow A.