Divisible by this year??? (Part 5: SAW)

I want to play a game. There are \(2015\) tiles. I want you to occupy the middlemost tile, the \(1008^{th}\) tile, and move on another tile in such a way there will always be \(2014\) unoccupied tiles. Please follow these steps:

  1. You must pick randomly a ball from a box. There are \(2014\) balls in the box numbered with integers \(1\), \(2\) , \(3\), …, \(2014\). let \(n\) be the number that you picked.

  2. If \(2014^2\) divides \(n!\), move one tile on your left. If not, move one tile on your right.

  3. Return the ball into the box and shuffle

  4. Repeat steps \(1\)-\(3\) until you are on the leftmost or the rightmost tile. If you are on the leftmost tile, you will lose and suffer but if you are on the rightmost tile, you will win and escape safely.

Based on this, the probability that you will win can be expressed as \(\frac{m}{n}\) where \(m\) and \(n\) are positive coprime integers. Find \(m + n\)


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