# Coin thrown 100 times. Find the probability that "coat" does not fall by more than 50 cases. Use the limit of the integral theorem of Laplace.

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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

`P_0+P_1+...+P_99+P_100`

`=(P_0+P_100)+(P_1+P_99)+...+(P_49+P_51)+P_50`

`=2(P_0+P_1+...+P_49)+P_50=1,`

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.` **

**Sources:**