@GT%20Wreck : There was no mention of the false negative (FN) rate, so it is assumed to be 0, not 5%. Ray’s answer was correct based on that assumption.
Assuming a 5% FN rate, your calculations are correct. Say 100,000 were tested. 4995 would test FP. 100 would be infected, but with a 5% FN rate only 95 would test positive. So the probability of infection when testing positive would be 95 / 5090 or 1.87%
The probability of infection when testing negative would be 5 / 94905 or .005%
There was no mention of the false negative rate, so it is assumed to be 0, not 5%.
The question said it was 95% accurate. That actually implies 5% false positive AND 5% false negative.
@GT%20Wreck: I now think you are right about the FP calculation. The 5% FP rate should be applied to the total uninfected (999), not the total tested (1000). But the answer comes out almost exactly the same either way (~1.96%), if we assume the FN rate is 0. Your mistake was assuming a 5% FN rate. Tests for disease try to keep the FN rate much lower than the FP, since a FN has more dire consequences to the patient than a FP.
On a side note, it would be interesting to know what the FP/FN rates are for the typical employee drug test, since an FP is much worse for the employee than a FN. This article says the FP rate can be as high as 10%, while the FN can be up to 15%.
@MikeInNH: Immediately after stating the 95% accuracy rate, Ray explained what that meant (5% FP out of 1000), so following those rules, he got the right answer. There’s no reason to assume the FN will equal the FP rate in any given test.
Randon clusters my boy, random clusters. When I go to the casino I think in terms of random clusters of events, positive or negative. So if I get four heads in a row (wins) I quit and move on. Even out of 240,000 positions on ye ole whirling wheel, the wins tend to be random but in clusters. Works for me.
, Ray explained what that meant (5% FP out of 1000), so following those rules, he got the right answer. There's no reason to assume the FN will equal the FP rate in any given test.
That’s bad on his part. You can’t assume there were no FN.
I always wondered why I hated math. Now I know. Thanks guys. I’m a person that like answers that are…not what might be.
Thanks guys. I'm a person that like answers that are....not what might be.
You’re definitely bad a math if you assume that. Math is very exact. There can be only ONE answer. There may be many paths to the answer…but the final MUST be the same or you did something wrong.
No…not bad at math. I used the hexadecimal system every day at work for 20 years+ and still do to some degree with computers. Split a dime 3 ways exactly and get back to me. That math will never be exact. I just hate math…there’s a difference.
Wow, there’s even a name for it: The False Positive Paradox. Pull quote:
Imagine running an HIV test on population A of 1000 persons, in which 40% are infected. The test has a false positive rate of 5% (0.05) and no false negative rate.
Most of my last post was to show what Ray’s answer would have to lead to and how IMPROBABLE his scenario really was. (NOT that I agreed with ANY of his approach.)
On the rare occasions when Ray is SOOOOO off base, and Tom agrees with him, I sure enjoy the FUN of gouging at their “goof up” sarcastically as long as I can. If a person doesn’t enjoy a good laugh by all means DON’T listen to Car Talk.
THE CORRECT ANSWER: 95% (or at minimum 19 out of 20) chance of those who tested positive.
AND .1% (or 1 chance in 1000) of those exposed to the disease.
From the original puzzler presentation:
Then you get a little bit encouraged. You say, “Wait a minute, doc, is this test 100 percent accurate?” Your doctor responds, “Well, not really. It’s 95 percent accurate.” In other words, 5 percent of the people who take the test will test positive but they don’t really have the disease."
The original question:
“What are the chances that you actually HAVE the disease?”
Other Thoughts
RAY’S TOWN CAN’T BE A "TOWN"
In “Ray’s Town” in my previous comment I consistently referred to his scenario as “your town”, his town’s population would have to had been of at lease 950,000.
No place could I find on the internet, anywhere in the united States, a population that big ever referred to as a “town”. Anything as large as 950,000 would be considered to be a “CITY”.
The largest population of any place considered a town in the US, I could find was Hempsted in New York with a population of 755,924 (and I know the puzzler does not limit us to the US).
UNDISPUTED FACTS ABOUT THE REAL PUZZLER TOWN:
The population of the actual puzzler town has to have AT THE LEAST 19,000, AND ALL OF THAT 19,000 WILL HAVE BEEN EXPOSED TO THE DISEASE. (There could be more but they would be ones not exposed or those who tested negative.)
WHY YOU ASK?
Out of all those who will test positive, 5% won’t actually contract the disease, and 95% of those tested positive WILL contract it.
The smallest number 5% can be is 1 person.
100% of those who tested positive would be a minimum of 20.
And 95% of them would be those that tested positive that WILL contract the disease and would be 19.
To maintain the 1 in 1000 (0r .1% ratio) of persons actually contracting the disease to all who were exposed, there must be 1000 times the number who actually will contract the disease. That calculates to 19 times 1000 which equals 19,000
Those who test Positive will have to be a multiple of 20 to have a whole number for 5%.
The 19,000 would have to be multiplied by the same factor.
.
Would you say a disease “SWEPT” through a town when only one out of every 1,000 people exposed contract the disease?
Still wrong. See False Positive Paradox above.
@MikeInNH: If the FN rate is unknown, the problem can’t be solved. Assuming it is equal to the FP rate is not justifiable, since in the real world that’s rarely the case. The only reasonable premise for purposes of solving this puzzler is that it is zero.