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.

 

[blue_raver22]

Hello,

Thank you for reaching out. Your question is an interesting one and involves the use of the inverse hypergeometric distribution. This distribution is used to calculate the probability of obtaining a certain number of successes in a sample, given a specific number of successes in the population. In your case, you are trying to find the number of blue balls (successes) in a population of 100 balls, in order to have a certain probability of obtaining 3 or more blue balls in a sample of 10 balls.

The general equation for the inverse hypergeometric distribution is:

Y = (K*N*R - N*K*X) / (R*X - K*R)

Where:
Y = number of successes in the population
K = sample size
N = desired number of successes in the sample
R = desired probability of success
X = population size

In your specific example, this would translate to:

Y = (10*3*0.85 - 3*10*100) / (0.85*100 - 10*0.85)
Y = (25.5 - 300) / (85 - 8.5)
Y = -274.5 / 76.5
Y = -3.59

This result is not a valid answer, as it implies a negative number of successes in the population. This could mean that your desired probability and sample size are not feasible with the given population size. Alternatively, it could also mean that there is an error in your calculation method.

I would suggest checking your calculations and making sure they are correct. If you continue to get a negative result, it may be helpful to consult with a statistician or seek further guidance on the specific problem you are trying to solve.

I hope this helps and good luck with your statistical calculator for game analysis. Let me know if you have any further questions.

Best regards,