Impossible Riddle?

 Say that there is a prison with exactly one-hundred prisoners in it. Now, the warden to this prison is extremely sadistic, and gives every prisoner a number. He then takes each number and puts them in a box, so we have one hundred boxes each with a number inside them. Then, he numbers each box, one to a hundred because that's how many boxes we have. He then tells the prisoners that each one can open exactly fifty boxes. If every prisoner finds their number, then they all get to go free, but if even one of them doesn't find it they all get executed. Now, the chances of them all finding their number by opening random boxes is 1/2 to the power of a 100, or roughly 7.9 x 10^-31. This is an incredibly small number. If the prisoners do it this way, then they don't stand a chance. However, there is a way to get their chances up to 31% instead of roughly 7.9 x 10^-31%. If they pick the box that is labeled with their number, and then go to the box labeled with the number inside the first box, their chances become (100!/100)/100!, which goes to 1/100. That's the chance of them finding their number by doing the above and making a loop containing every single box. Using the above math, we can prove that the chances of their being a loop like this longer then is 1/51, which puts the chances of failure at about 69%, and the chances fo success at 31%.