Hyper-geometric Distribution

Date: 07/29/99 at 08:59:47
From: Evette
Subject: A Probability Problem

Through a mix-up on a production line, 6 defective refrigerators were 
shipped with 44 good ones. If five refrigerators are selected at 
random,

  a. What is the probability that all five are defective?

  b. What is the probability that at least one is defective?

  c. What is the probability that no more than one is defective?

Date: 07/29/99 at 10:28:24
From: Doctor Anthony
Subject: Re: A Probability Problem

This type of problem is known as hyper-geometric distribution.  

I shall be using the C(n, r) notation as described below.

C(n, r) Explanation
-----------------------------
Use the notation C(n, r) to mean the number of combinations of r 
things that can be made from n different things. The formula for C(n, 
r) is:

                n!                                 10!
  C(n, r) = --------    so   C(10, 4) = ---------  = 210
             r!(n - r)!                            4! 6!

Now, answering your first question,

There are 50 items, out of which 44 are okay and 6 are defective. A 
sample of 5 is examined.

                       C(6, 5)          6
 Prob(5 defective) =  ---------  =  ---------  = 0.000002831845
                       C(50, 5)      2118760

Answering your second question,

 Prob(at least 1 defective) =  1 - P(none defective)

                                    C(44, 5)          1086008
                            =  1 - ----------  = 1 - ---------
                                    C(50, 5)          2118760

                            =  0.48743

Answering your third question,

                             C(44, 5)     C(6, 1) x C(44, 4)
  We require P(0) + P(1) =  ---------- + --------------------
                             C(50, 5)          C(50, 5)

                        =  0.896993

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/