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### Checking Account Balance

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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/
```
Associated Topics:
High School Probability

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