Checking Account Balance

Date: 05/08/97 at 19:49:48
From: Lisa Grove
Subject: Math Problem

My math class had the problem below for homework and all of us 
(including my math teacher) got an answer different from the textbook. 
If you could please let us know what you think the answer is and how 
you got it, we would be very grateful.  Here's the problem: 

Suppose your mom and dad have a joint checking account with a total of 
$400. You have talked each of them, unkown to either of them, into 
writing you a check for a random amount less than $400. 

What is the probability that more than $80 remains in the account 
after the checks are written?

Thank you!
Sarah Grove

Date: 05/09/97 at 09:12:50
From: Doctor Anthony
Subject: Re: Math Problem

Dear Sarah,

You will need to make an assumption on the probability distribution of 
the sum of money that you persuade them to give you. I will assume a 
rectangular distribution. This means that each parent is equally 
likely to give you any sum between $0 and $400.

This type of probability problem is best done by graphical means.  
We assume x is the amount given you by your father and y is amount 
given by your mother.  x and y are both distributed uniformly between 
0 and 400.  If you represent x and y on the usual rectangular axes, 
then you can be positioned with equal probability anywhere in the 
square with corners at (0,0), (400,0), (400,400), and (0,400).  

Now if we have more than $80 left in the account, it means x+y < 320.  
So on the axes already drawn, we plot the line x+y = 320. We must lie 
to the left of this line, on the side which includes the origin, 
because this represents all possible check book balances above $80.

The probability of being in this region is the area of the triangle 
formed by the x and y axes and the line x+y = 320 divided by the area 
of the square of side 400.

The area of triangle = (1/2) x 320 x 320.

The area of square = 400 x 400.

                        (1/2) 320 x 320
Required probability =  ----------------  = 0.32
                           400 x 400

So the probability that more than $80 remains in the account is 0.32

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/