# Basic problem

• Sep 19th 2010, 11:50 AM
sfspitfire23
Basic problem
There are 12 governor seats up for reelection. there are only 2 candidates in each race--democrats and republicans and the each have an equal shot at winning.
What is the probability that half of the seats are won by democrats and half are won by republicans?

Work:
Each candidate has a 1/2 chance of winning or losing....but I can't seem to line this answer up.
• Sep 19th 2010, 11:58 AM
CaptainBlack
Quote:

Originally Posted by sfspitfire23
There are 12 governor seats up for reelection. there are only 2 candidates in each race--democrats and republicans and the each have an equal shot at winning.
What is the probability that half of the seats are won by democrats and half are won by republicans?

Work:
Each candidate has a 1/2 chance of winning or losing....but I can't seem to line this answer up.

The number won by the the Republican has a binomial distribution B(0.5,12), and you are asked for the probability of them wining exactly 6, which is b(6;0.5,12)

CB
• Sep 19th 2010, 12:03 PM
sfspitfire23
Is there a way you could calculate this by plugging numbers in for the 12 spaces? So for example, if the question asked for the probability of democrats winning all 12, the answer would be (1/2)^12.
• Sep 19th 2010, 06:58 PM
CaptainBlack
Quote:

Originally Posted by sfspitfire23
Is there a way you could calculate this by plugging numbers in for the 12 spaces? So for example, if the question asked for the probability of democrats winning all 12, the answer would be (1/2)^12.

That is unclear, of course I can calculate this, probably with my eyes shut. What I would like is for you to be able to calculate it. The formula for binomial probabilities is:

$b(n,p,N)=\dfrac{N!}{n!(N-n)!} p^n(1-p)^{N-n}$

Look up the binomial distribution on Wikipedia. It will tell you everything you need to know about the binomial distribution.

CB