Most likely of two examples

Hi My instructor gave us two examples and posed a question:

We have a fair coin which is more likely?

1. We flip the coin 100 times and see exactly 50 heads.

2. We flip the coin 1000 times and see exactly 500 heads.

Someone in the class said immediately that 1 is more likely by a factor of sqrt(10). Why is sqrt(10) the difference? Now explanation was given.

Re: Most likely of two examples

Hey mathplease.

Hint: Try calculating the actual probability using the PDF (or by using a Normal distribution approximation with continuity correction).

Have you come across using the Normal distribution to approximate a binomial distribution for large values of n?

Re: Most likely of two examples

The binomial distribution with N= 100, p= 1/2 has mean 50 and standard deviation $\displaystyle \sqrt{(100)(1/2)(1/2)}= \sqrt{25}= 5$. To approximate that with the Normal distribution, use a normal distribution with that mean and standard deviation and find the probability that x is between 50- 1/2= 49.5 and 50+ 1/2= 50.5.

To do the same with N= 1000, p= 1/2, mean is 500 and standard deviation $\displaystyle \sqrt{1000}(1/2)(1/2)}= \sqrt{250}= 5\sqrt{10}$. Find the probability of x between 500- 1/2= 499.5 and 500+ 1/2= 500.5.