Law of large numbers application

Question

I came around this question on the web:

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 Law Of Large Numbers? How should the statement be positioned for Law Of Large Numbers to work?

Am I right to conclude that expected number of boys are 25 and 250 respectively in the hospitals? Also I don't seem to understand the application of law of large numbers here, can someone shed light on this issue?

Answer 1

The Law of Large Numbers tells us that the Average behavior tends to the Expected Value as the sample size increases, but it doesn't help you compute exact values. Indeed, for large samples, the probability of getting any exactly specified ratio tends to $0$. In this case we can compute everything precisely.

This is a binomial process, with $2n$ trials and $p=\frac 12$. Specifically, you are interested in $n=25,250$. We see that the probability of a tie is $$\psi(n)=\binom {2n}n\times \frac 1{2^{2n}}$$

The numbers are small enough to compute both. We get $$\psi(25)=0.112275173,\;\psi(250)=0.035664646$$

Answer 2

The 50-bed hospital is far more likely to have equal numbers of boys and girls.

The law of large numbers tells us that the proportion of boys is likely to approach 0.5 more and more closely as the number of beds becomes larger and larger. By that, I mean that $actual \> proportion\> of \>boys \rightarrow 0.5$, or in other words, $\frac{actual \> proportion\> of \>boys}{0.5}\rightarrow 1$. It tells us nothing about the probability that the proportion of boys is exactly 0.5, which falls as the number of beds increases. It doesn't even tell us that the expected difference between the number of boys and half of the total number of babies decreases. In fact that expected difference increases.

You are right to think that the expected proportion of boys is 0.5 in both cases. Indeed it would be 0.5 even if the number of beds were odd.