Finding the Probability of Flipping 500 Heads and Tails

[bmb2009]

Homework Statement

Suppose you flip 1000 coins, what is the probability of flipping exactly 500 heads and 500 tails. (Hint: First write down the formula for the total number of possible outcomes. Then, to determine the "multiplicity" of the 500-500 macro state, use stirlings approximation.

Homework Equations

The Attempt at a Solution

I have done a couple of these problems (successfully) with the binomial distribution method so am I wrong in stating the following equality? Or is this correct and I should just use the stirlings method to make N!~ NlnN - N at large N and then use this to simplify the probability I wrote? Thanks!

probability = (1000 choose 500)((.5)^500)*((.5)^500)

 

[mfb]

It is correct. Stirling's formula will allow you to calculate that, yes.
And WolframAlpha can do it as well.

 

[Ray Vickson]

Homework Statement

Suppose you flip 1000 coins, what is the probability of flipping exactly 500 heads and 500 tails. (Hint: First write down the formula for the total number of possible outcomes. Then, to determine the "multiplicity" of the 500-500 macro state, use stirlings approximation.

Homework Equations

The Attempt at a Solution

I have done a couple of these problems (successfully) with the binomial distribution method so am I wrong in stating the following equality? Or is this correct and I should just use the stirlings method to make N!~ NlnN - N at large N and then use this to simplify the probability I wrote? Thanks!

probability = (1000 choose 500)((.5)^500)*((.5)^500)

Yes, that is correct. Notice also that $.5^{500} \times .5^{500} = .5^{1000} = 1/2^{1000}$.