# Most likely of two examples

• Aug 27th 2013, 10:05 PM
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.
• Aug 27th 2013, 10:49 PM
chiro
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.