Lung Cancer and Smoking

Date: 10/20/2001 at 22:57:08
From: nin
Subject: Probability

Medical researchers know that the probability of getting lung cancer 
if a person smokes is .34. The probability that a nonsmoker will get 
lung cancer is .03.  It is also known that 11% of the population 
smokes. What is the probability that a person with lung cancer will 
have been a smoker?

Date: 10/21/2001 at 05:36:52
From: Doctor Mitteldorf
Subject: Re: Probability

Good question. Here's a clear way to think about it:

Start by dividing the population into the 89% that doesn't smoke and 
the 11% that does.

Now subdivide further: 89% times 0.03 is the probability of not 
smoking and getting lung cancer. 89% times 0.97 is the probability of 
not smoking and NOT getting lung cancer, etc.  You can make a chart 
this way:

                 Cancer     No cancer
Smokers        |   3.74% |  7.26%   |
Non-smokers    |   2.67% | 86.33%   |
                ---------  ---------
Totals             6.41%   93.59%

The question asks just about people with cancer, so our universe is 
just the first column. We can "re-scale" the probabilities to 100% by 
dividing both numbers by 6.41%, which is the total for the column:

If a person has lung cancer, he has a probability 3.74/6.41 = 0.58 
of being a smoker, and a probability 2.67/6.41 = 0.42 of being a 
non-smoker.

(In the real world, I'm afraid more than 11% of people smoke, or have 
smoked, by the time they get to the age where lung cancer is
tragically common. So in real life, a much larger majority of lung 
cancer patients are smokers or former smokers.)

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/