Simplifying a Probability Question

Date: 08/19/2008 at 08:11:04
From: Aileen
Subject: sat

A computer software firm maintains a telephone hotline service for its
customers.  The firm finds that 48% of the calls involve questions
about the application of the software, 38% involve issues of
incompatibility with hardware and 14% involve the inability to install
the software on user’s machine.  These three categories of problems
can be resolved with probabilities 0.90, 0.15, and 0.80 respectively.

(1) Find the probability that a call to the hotline involves a problem
that cannot be resolved.

(2) Suppose that a call to the hotline involves a problem that cannot
be resolved.  What is the probability that it is an issue of
incompatibility with the hardware?

                                             
Date: 08/19/2008 at 08:47:37
From: Doctor Ian
Subject: Re: sat

Hi Aileen,

Note that you can eliminate a lot of irrelevant detail by changing the
problem to one involving pulling colored and numbered balls from a jar.  

For example, suppose we have a jar with 1000 balls:

  480 are azure      (these represent application questions)
  380 are crimson    (these represent compatibility questions)
  140 are indigo     (these represent installation questions)

Of the 480 azure balls, 90 percent, or 432, are numbered with a 1,
representing questions that can be resolved; the the other 48 are
numbered with a zero.  You can find similar ratios for the other two
colors. 

So now, the question 

  Find the probability that a call to the hotline 
  involves a problem that cannot be resolved.

is equivalent to the question

  What is the probability of pulling out a ball with a zero
  on it?

and the question

  Suppose that a call to the hotline involves a problem that cannot
  be resolved.  What is the probability that it is an issue of 
  incompatibility with the hardware?

is equivalent to the question

  If I pull out a ball with a zero on it, what is the probability
  that it is crimson?

Can you see why the questions are equivalent?  Can you answer them in
this form?

Try that, and show me the steps you take. 

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

Date: 08/19/2008 at 18:40:11
From: Aileen
Subject: Thank you (sat)

Thank you a lot you are a great helper.  That made the problem a lot
easier.