*simple* probability question

B

buriboi1

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Dec 16, 2022
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1st post ever. You will be able to figure out that I don't know anything about math quite quickly, lol.

Let's say a lab test can return positive or negative, and the test is 55% accurate.

I do the test 10x, each time it returns positive. What is the formula to determine the overall likelihood that the positive is true?

(Sorry for my lack of math lingo!)
 
buriboi1 said:
1st post ever. You will be able to figure out that I don't know anything about math quite quickly, lol.

Let's say a lab test can return positive or negative, and the test is 55% accurate.

I do the test 10x, each time it returns positive. What is the formula to determine the overall likelihood that the positive is true?

(Sorry for my lack of math lingo!)
Click to expand...
There isn't enough information to determine.
 
buriboi1 said:
and why is that? I'm actually surprised by that answer!
Click to expand...
This type of question is typically summarized in what we call a Confusion Matrix.
Screen Shot 2022-12-23 at 8.35.56 PM.png

You're given P(accuracy) = P(True Positive) + P(True Negative) = 0.55
Furthermore, P(Positive) = P(True Positive) + P(False Positive) = 1.0

Mathematically speaking you have a system of 2 equations and 3 unknowns. Whenever we have a system of equations that has more unknowns than number equations, it can't be determined the exact value of each unknown.
 
BigBeachBanana said:
This type of question is typically summarized in what we call a Confusion Matrix.
View attachment 34674

You're given P(accuracy) = P(True Positive) + P(True Negative) = 0.55
Furthermore, P(Positive) = P(True Positive) + P(False Positive) = 1.0

Mathematically speaking you have a system of 2 equations and 3 unknowns. Whenever we have a system of equations that has more unknowns than number equations, it can't be determined the exact value of each unknown.
Click to expand...
Excellent. Thanks for the answer, it's appreciated.
 
To build on BBB’s answer, you could approximately answer your question ONLY if it is safe to assume the failure rate of the test is approximately the same for positive and negative results.

Suppose it is a test for the early detection of pancreatic cancer. Suppose it says the patient has cancer when that is true 15% of the time, and also says the patient does not have cancer when that is true 95% of the time. On average, the test seems to be 55% reliable (average of 15% and 95%), but it fails to detect actual cancer 85% of the time. When a test is accurate may matter.

It is the old story: averages may be very deceiving.

Think about that the next time you see that ad on television for home testing against colon cancer.
 
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