Law of Large Numbers Failed
Problem: There are two maternity hospitals in a town with 50 and 500 beds. Given full occupancy on a particular day, which of these hospitals is more likely to have equal no of boys and girls given probability of boys = probability of girls ? What would the answer intuitively be by #LawOfLargeNumbers? You would see #LawOfLargeNumbers does not seem to work here. How should the statement be positioned for #LawOfLargeNumbers to work? Solution: Highlight the part between the * symbols for the answer. * The SMALLER (50-bed) hospital is more likely to have an exactly equal split. P(exactly 25 boys in 50) = C(50, 25) / 2^50 = 11.2%; P(exactly 250 in 500) = C(500, 250) / 2^500 = 3.6%. The exact-split probability falls like 1/sqrt(pi*n) (Stirling), so it shrinks as the hospital grows. No contradiction with the Law of Large Numbers: the LLN is a statement about PROPORTIONS - P(|boys/n - 1/2| < epsilon) -> 1 for any fixed epsilon - not about hitting exact equality. The f...