Are They Wearing Seatbelts?

Date: 26 Mar 1995 06:27:58 -0500
From: Anonymous
Subject: Seatbelts

80% of all California drivers wear seatbelts.  If 4 
drivers were pulled over, what is the probability that all 
4 would be wearing their seat belts?  What is it that none 
of them would be wearing their seatbelts?
Thanx

Date: 26 Mar 1995 14:10:04 -0500
From: Elizabeth Weber
Subject: Re: Seatbelts

Hello there!

The number of drivers in California is large enough that 
pulling one over won't affect the % left who are wearing 
seatbelts.  This makes your problem pretty straightforward.  

The probability that any driver we pull over is wearing a 
seatbelt is 80%

The probability that any second driver we pull over is also 
wearing a seatbelt is also 80%, so the probability that both 
of the drivers we pull over are wearing seat belts is 80% of 
80%, or .8 x .8.

To see why this is so, you can try drawing something called 
a Venn diagram.  Take a piece of paper, and mark it so that 
it's divided into two pieces, of which one takes up 80% of 
the paper, and the other takes up 20%, something like this:

 __________________
|       20%        |   --probability that the driver
|__________________|     is not wearing a seat belt
|                  |
|                  |
|                  |
|       80%        |  --probability that the driver
|                  |    is wearing a seat belt
|                  |
|                  |
|__________________|

This is how the probability of pulling over a driver wearing 
a seatbelt looks after you've pulled over one driver.  Now, 
let's say you've pulled over a driver wearing a seatbelt, and 
you want to see what the probability of pulling over a second 
driver who is wearing a seatbelt looks like.  You're already 
inside the piece of your paper which takes up 80% of the 
paper, so you divide this piece into 2 smaller pieces, just 
like you divided the entire piece of paper before:

 __________________
|       20%        |
|__________________|
|       |          |
|       |          |
|    80%|          |     
|     x | 80% x 80%|
|    20%|          |
|       |          |
|       |          |
|_______|__________|
              
The 80% x 80% box is the box you're interested in.  Now, can 
you see how you would divide it to find the probability that 
the third driver you pull over is wearing a seatbelt if the 
first two you've pulled over were wearing seatbelts?  Can 
you see how to find the probability of pulling over 4 drivers 
in a row who are all wearing seatbelts?

Can you see how to find the probability that none of the 
drivers you pull over are wearing seatbelts?

If you need more help, just let us know.

                        Thanks for writing to Dr. Math!
                                Elizabeth, a math doctor