Statistics: Polls and probability

[Niles]

Homework Statement

Hi all.

At a poll in America, where 1000 people have been asked, 480 people said that they would vote yes on proposition A (it is a fictive proposition), and of course 520 people said they would vote no on proposition A.

I have to find the probability that proposition A will pass.

What I have done is to use the following formula:

$$P = \frac{{(n + 1)!}}{{m!(n - m)!}}f^m (1 - f)^{n - m},$$

which has been derived in class. Here m is 480, n is 1000. P is the probability of M persons answering "yes" out of N total (so N is the total amount of people in America voting, and M is the amount in America that votes "yes") when we have a sample of n (1000), where m (480) have answered "yes". f is equal to M/N.

My attempt is that if more than 50% vote yes, then prop. A will pass. So I want to integrate P from N/2 to N, but it is not a continuous distribution. I thought of finding P for N/2 (i.e. that 50% will answer "yes"), but that is only the probability of 50% answering yes, and does not include the rest from > 50%.

What would be the most proper approach to this problem?

Thanks in advance.

 

[statdad]

The proposition will pass if at least 50% of the voters vote for it, so your basic idea is correct.
Try the normal approximation to find the probability you need.

 

[Niles]

Hi statdad

Thanks for replying. The normal approximation gives us:

$$P \propto \exp \left( { - n\frac{{(f - f_0 )^2 }}{{2f_0 (1 - f_0 )}}} \right),$$

where f₀ is the mean, i.e. m/n. This is still a discrete distribution, so what is the next step from here?

Thanks in advance.