I can answer this, but it doesn't require the Central Limit Theorem, which is what I think you mean by Laplace's Theorem? Since this has been unanswered for a while, I'll show you how I would do it.
There are 101 possibilities for the number of "coats": 0 coats, 1 coat,...,100 coats. These are not equally likely, but there are pairs that are equally likely. More specifically, the probability of 0 coats is the same as 100 coats, the probability of 1 coat is the same as 99 coats, etc. These all add to 1. If we let `P_n` be the probability of `n` coats, then we have
and we wish to find `P_0+P_1+...+P_50.` We can get `P_50` fairly easily by calculating `_100 C_50 (1/2)^100~~0.080.`
Then we solve for `P_0+...+P_49~~0.92/2=0.46.` Thus the probability of getting no more than 50 coats is approximately
`0.54,` and the exact answer is found by keeping the exact answer for `P_50` instead of using `0.08.`