Friday, October 25, 2013

surf's up... the solution

I think it's easy for people that love math and have studied it for years to quickly grasp mathematical ideas and solutions, simply by reading them.  To those with passions in other disciplines, reading about how to do a math problem may be both unsatisfying and a challenge.  I am going to try to explain the solution to the surfing problem without any fancy math terms or formulas.  The answer will be a little less precise, but I hope to give you some insights in how to think about the problem  You can tell me if this makes sense.

Here's the question again: 

The Surfboard Store sells Special Surfboards that
are so carefully made that only 1 in 1 thousand
is bad. The store tests all Special Surfboards
using a test that is 99% accurate. If you buy
a Special Surfboard that tests bad, what is the
probability that it really is bad?

Start with the first sentence:  One in a thousand surfboards is bad.  Let's look at a thousand surfboards.  How many are actually bad?  I hope you said 1. 
The next sentence says that the test is 99% accurate, meaning that 1% of the time it misidentifies a surfboard....  so in the remaining 999 good surfboards, 1%  or about 10 will be identified incorrectly as bad when they are really good. 
Thus we have 11 surfboards identified as bad, but there is only one surfboard that is actually bad in the bad group.  1/11 is roughly .09 or 9%.
Thus the probability that a surfboard identified by the 99% accurate test as bad actually is bad is only about 9%. 
What makes this idea important in my mind is less about the calculations and more about the counter intuitive nature of the solution.  When we hear that a test is 99% accurate, we tend to think that it must be absolutely correct.  

There are interesting implications when we apply this idea to politics.  There are life-changing implications when we consider this idea in the context of medicine. 

Statistics is an incredibly important and largely misunderstood discipline.  Math teachers have an important responsibility in addressing this problem.  I'm not sure we are doing the job very well. 

I welcome your comments on this problem's explanation and on statistics in general.

1 comment:

  1. Thanks so much! It makes a lot senses to me!!!