Prisoner

There are one hundred extremely smart prisoners that are in one hundred solitary prison cells. There are also one hundred boxes on the inside of the cells. Each box contains a name of a prisoner. Each box has a slip of paper that is only in one box and in one of the hundred boxes! The warden gives a game to the prisoners and if prisoners win they will get free but if they lose, they will get killed. The warden decides that the first prisoner can open every box and switch and name they want. If all the prisoners get their name, they will get set free but if even one prisoner does not get his name, all the prisoners will get killed. The prisoners make a plan. One week later, the first prisoner goes out of his cell and switches around the names. For example, like if prisoner one looks inside box one and he finds thirty-two, he will put the paper into box thirty-two and if he finds seventy-five in that box, he will put the paper into box seventy-two and etc. What is the chance the prisoners will win the game? $$