Cancer Problem explanation

[Nathan Foley]

Hey all,

I really enjoyed the cancer problem from this week's class. I
discussed it with a few friends of mine, none of whom found the puzzle
as interesting as I did and all of whom struggled to follow my
explanation of the solution. I ended up drafting an email
walkthrough, and I thought it'd be interesting to share it here.

Enjoy!

-------------

First, the puzzle (in case anyone here didn't read it):

There's a 1% chance you have cancer. You are given a test with the
following limitations:

* If you HAVE the cancer, it is 10% likely to return a false negative,
* If you DO NOT HAVE the cancer, it is 20% likely to return a false
positive

The test is administered twice. Both results are positive. What is
the likelihood that the tests are accurate?

-------------

Let’s start with the base information. We’re given three data points,
which I expand into six:

1% of having the cancer
99% of not having the cancer
90% for a positive result on cancer positive patient
10% for a negative result on cancer positive patient
20% for a positive result on a cancer negative patient
80% for a negative result on a cancer negative patient

We’re given two positive results.

Now we look at Test One’s outcome by itself for a moment. Here’s the
full breakdown:

Cancer/Test
• Positive/Positive = 1% * 90% = 0.9%
• Positive/Negative = 1% * 10% = 0.1%
• Negative/Positive = 99% * 20% = 19.8%
• Negative/Negative = 99% * 80% = 79.2%

Overall, that translates to a 20.7% chance of a positive result and a
79.3% chance of a negative result. We’re only interested in positive
results, so we throw out the negative ones leaving us with 20.7% (only
0.9% of which are accurate). So the likelihood of an accurate
positive outcome after one test is 9/207, which can be reduced to
1/23. Roughly 4.35%.

This is the point at which I made my mistake. Once I had the 1/23 I
looked at it and said, “okay, now I just need to do it again.” I know
I’m getting a positive result in Test Two, and I know it’s a 1/23 of
being “successful”, so I look at the likelihood of “failure” (22/23)
and square it:

22/23 * 22/23 = 484/529

If 484/529 is the chance of getting two false positives, then 45/529
is the chance that at least one of the tests was accurate. Which sort
of makes sense, right? The percentage is going from 4.35% to 8.51%,
which seems reasonable. Except for one problem.

You either have the cancer or you don’t; that condition can’t change
from one test to the next. So while we have four possibilities for
the first “roll” (the Patient/Test chart above), the second test is
limited by the condition(s) we picked for the first test. This gives
us three pieces of information at the start of Test Two:

• The positive result from Test One is accurate in 1 case and
inaccurate in 22,
• The result from Test Two will be positive,
• Regardless of the test results, we can only be in one state at a
time - actually positive or actually negative (unless you’re
Shrodinger’s cancer patient). That is, if you got a false positive in
Test One then you can’t get a true positive in Test Two, and vice
versa.

That simplifies the hell out of Test Two (particularly since we get to
throw out all the negative results). Here it is:

Test One was positive (1 in 23): Test Two is 90% likely to return a
positive result [1 * 0.9 = 0.9]
Test One was negative (22 in 23): Test Two is 20% likely to return a
positive result [22 * 0.2 = 4.4]

Plugging those in we get 9:44, or 9/53. The likelihood of cancer has
jumped from 4.34% to 16.98%.

[Nathan Foley]

Minor correction to that final bit:

Test One was a true positive (1 in 23): Test Two is 90% likely to

return a
positive result [1 * 0.9 = 0.9]

Test One was a false positive (22 in 23): Test Two is 20% likely to

[camilo cervantes]

Thank you, i missed that one and now i understood

2011/10/26 Nathan Foley <nrf...@gmail.com>

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Camilo Cervantes S.
Est. Ingeniería de Sistemas
Universidad de Córdoba
Grupo de desarrollo 64 bits
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