Geometric Probability

Date: 5/1/96 at 2:20:0
From: Anonymous
Subject: Geometric Probability

About once a year an asteroid enters the solar system in the region
between the sun and the orbit of the planet Mars.  What is the 
probability that it will hit the earth?  The distance from the sun to 
Mars is 142 million miles.  The diameter of the earth is 7927 miles, 
and the diameter of the sun is 864,400 miles.

Using the formula for area of a circle we found the area of the entire
circle from the center of the sun to Mars.  Then we subtracted the 
area of the sun, calculated the area of the earth, and set up a 
ratio... but came nowhere near the answer on the answer sheet of 
7.7436x10-10 or 7.7908x10-10.  Where did we go wrong?  We're going 
crazy here.

Please give us as detailed an answer as possible about the process 
involved in solving this problem.  Mom's losing sleep!

Date: 10/31/96 at 12:7:43
From: Doctor Ceeks
Subject: Re: Geometric Probability

Hi,

A simple measure of the probability would be the ratio of the earth's 
cross-section divided by the area of the orbit of Mars:

      cross-section of earth
  p = ----------------------
        area of Mars orbit


      pi * (7927/2)^2
    = --------------------
      pi * (142,000,000)^2


          7927/2
    = ( ----------- )^2
        142,000,000

When I evaluate this, I get 7.8 * 10^-10. 

When you computed the 'area of the earth', did you perhaps compute the 
surface area of the earth, instead of the cross-sectional area? 

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