Answer:
Given that:
A fair coin is tossed n times, coming up Heads NH times and Tails times. Let
Use Cramer’s Theorem to show that for 0 < a < 1,
limn→∞
Let the number of heads be x times, then the number of tails will be n-x. The variable Sn then becomes x-(n-x) or 2x-n.
Now the probability of getting x heads in n tosses is nCx*(1/2)^x*(1/2)^(n-x) or nCx*(1/2)^n. and P(x>a) for a is a integer will be summation nCx*(1/2)^n from a+1 to n.
Thus for the expression P(S>a) can be written as
2x-n>a or x>(a+n)/2, thus the probability becomes summation nCx*(1/2)^n from integer((a+n)/2) to n.as a is a fraction less than 1, thus the int(a+n)/2) becomes simply int(n/2) or half the series, thus the summation is simply 2^(n-1).
Thus the expression becomes 1/2n
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