Nervous sharp shooter
A basketball player is practicing free throws in the gym. He makes the first throw in, but misses the second throw. At this point his nerves kick in and he is unable to make the shots consistently. The probability that he makes any subsequent throw successfully is the percentage of prior throws that he makes in. (E.g. If he has successfully made 3 throws out of 5, the probability that he will make the 6th throw is 3/5).
His coach knows this probability and walks off the court after the second shot. He comes back and sees the player make his 2015th shot in. From the coaches point of view what is the probability that the player makes the 2016th shot in?
Write the probability as \(\frac ab\) where \(a\) and \(b\) are coprime. Enter \(a+b\) as your answer.