Using Tables to Find Probabilities

Date: 10/23/2003 at 21:35:35
From: Dominic
Subject: second grade probability problem

Levi has 3 blue shirts and 1 red shirt.  He has 1 pair of white slacks 
and 1 pair of blue slacks.  The probability that he will wear white
slacks and a blue shirt is ________ out of _________.

I would think that you have 

  BS BS BS         RS
     WS            BS

Is the answer 3 out of 4?

Date: 10/25/2003 at 12:11:44
From: Doctor Ian
Subject: Re: second grade probability problem

Hi Dominic,

Here's one way to think about it.  Let's make a table, with the slacks
along one side, and the shirts along the other:

                          slacks
                   white           blue

            blue     ?               ?

   shirts   blue     ?               ?
 
            blue     ?               ?

            red      ?               ?

In each location of the table, we can write down what combination he's
got, if he chooses the corresponding colors.  For example, if he
chooses a red shirt and white slacks, we get

                          slacks
                   white           blue

            blue     ?               ?

   shirts   blue     ?               ?
 
            blue     ?               ?

            red     RW               ?

Does that make sense?   Filling in the table, 

                          slacks
                   white           blue

            blue    BW              BB

   shirts   blue    BW              BB
 
            blue    BW              BB

            red     RW              RB

So there are 8 possible ways that things can go, some of which look
the same.  (For example, there are 3 ways that he might choose a blue
shirt and white slacks.) 

The probability of a thing happening is defined this way:

                The number of ways the thing could happen
  probability = ---------------------------------------------
                The number of ways that anything could happen

We know that there are 8 possible ways for anything to happen.  And
there are 3 possible ways for him to end up with a blue shirt and
white slacks.  So the probability of choosing a blue shirt and white
slacks is 

                3
  probability = -
                8

(Note that this assumes he's going to choose randomly, e.g., by
grabbing items without looking for them.)

What's confusing about this is that if you just list the combinations
that look different, there are only four:

  BW:  blue shirt, white slacks
  BB:  blue shirt, blue slacks
  RW:  red shirt,  white slacks
  RB:  red shirt,  blue slacks

So it would be easy to think that the probability of BW should be 1 
out of 4.  But that's why we make the table--to take into account that 
the numbers of items of different colors aren't the same.  

To see why this is important, imagine that he's got one blue shirt, 
and a million red shirts.  Surely the probability of ending up with a
blue shirt isn't going to be 1 in 4!

Does this help? 

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