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### Using Tables to Find Probabilities

```Date: 10/23/2003 at 21:35:35
From: Dominic

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.)

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/
```
Associated Topics:
Middle School Probability

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