For a Given a Probability of Success, How Many Successes in a Sample?

[OsoMoore]

Is there a general equation for an inverse hypergeometric distribution?

Greetings,

I'm making a statistical calculator for game analysis, and have an interesting problem.

Here is a specific example: I have a sack of 100 balls and are going to take 10 out of it randomly. I want to find 3 or more blue balls 85% of the time.

How many blue balls must I have in the sack to have this probability of success?
X = 100
Y = ?
K = 10
R = 85%
N = 3

In general terms:
To achieve at least N successes R percent of the time in a sample size of K, how many items Y in a set of X items must be successes?

I arrived at a general equation by my own calculations to find N given R, but it indicates I need about 43 blue balls get achieve my desired 85% chance.

However, using a standard cumulative hypergeometric distribution to find R given N, I calculate that using 43 balls will give me a 88.9% chance of success.

Consequently, I know my method is in error, and am hoping I can have some help in figuring a general equation for this problem. It would seem what I am seeking is an inverse hypergemoetric distribution.

The equations and work so far can be seen in the google docs spreadsheet I have created for them http://spreadsheets.google.com/ccc?key=0AnPw5qvi2hRrdHoweklSQnBuVW9NbVFIUENpYmUyV3c&hl=en".
You can see my work on the Chance to Draw Stats and Probability tabs.

Any input would be appreciated.

 

[Mandark]

Having 40 blue balls gives you a 0.846ish chance of having 3 or more blue balls. In general the problem reduces to solving a polynomial equation. Do you know how to set up the equation?

 

[zli034]

Classic binomial distribution.

 

[zli034]

Classic binomial distribution.

I was wrong. It's a hypergeometric distribution problem. I have calculated if you want 84.6% probability to get 3 or more blue balls from random picked 10 balls out of 100 balls, the 100 balls need to have at least 40 blue balls and also 60 other colored balls.