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

  1. Feb 17, 2010 #1
    Is there a general equation for an inverse hypergeometric distribution?


    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.
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
  3. Feb 20, 2010 #2
    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?
  4. Feb 23, 2010 #3
    Classic binomial distribution.
  5. Feb 23, 2010 #4
    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.
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