What is the Probability of 5 People in a Small Town Getting Cancer in One Year?

[csnsc14320]

Feeling a bit rusty on my probability -

Say you have 70,000,000 people in a country and a small town with 13,700 people.

Now, say 3,000 - 5,000 people of the 70,000,000 people become infected with cancer per year.

What are the chances 5 people from the small town get infected in one year?

This isn't for a class so I welcome the right answer.

I was thinking something along the lines of

(3,000/70,000,000 * 13,700/70,000,000)^5 percent chance to(5,000/70,000,000 * 13,700/70,000,000)^5 percent chance, but I'm can't recall at the moment.

 

[bucher]

I don't think that this is the right way.

This looks like the probability of randomly choosing five people in the country and picking all five in the town and all five have cancer.

I remember doing some problems like these and they use factorials in them. I'm trying to think of the exact equation but it doesn't come to mind.

 

[csnsc14320]

I don't think that this is the right way.

This looks like the probability of randomly choosing five people in the country and picking all five in the town and all five have cancer.

I remember doing some problems like these and they use factorials in them. I'm trying to think of the exact equation but it doesn't come to mind.

Yeah, I guess in words what I am trying to do is find the probability of randomly choosing 3000-5000 people from 70,000,000 that exactly 5 of them would be from my town of 13,700 people

 

[CRGreathouse]

Yeah, I guess in words what I am trying to do is find the probability of randomly choosing 3000-5000 people from 70,000,000 that exactly 5 of them would be from my town of 13,700 people

Let's say that every person in your country has a p = 1 - q = 4000/70 000 000 chance of getting cancer. Then the chance that exactly k people get cancer in a town of 13,700 is p^k q^(13 700 - k) (13 700 choose k), which is small (0.11%) for k = 5.

But then again if there are 2000 such villages, the chance that at least one has 5 or more cases of cancer is 92%.

 

[ibcnunabit]

Nevermind, CRGreathouse got it. (p=0.0011195)

For csnc: the factorials come from the (n choose x) part of the formula. That would be defined as n! / (x!*(n-x)!) but nCr function on calculator works great. ;)
That substitutes into

p(x)=(n choose x) * p^x * q^(n-x)

,where

p=probability of "success" (sick person) =4000/70,000,000 (assuming we reduce the range to a single value and distribution is approximately symmetric),
q=1-p=probability of "failure" (well person),
n=number of trials (13,700), and
x=# of successes in n trials (looking for 5)