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Thank you for explaining that, I was genuinely curious about how the odds would increase, but I now understand that running the same test twice wouldn't necessarily improve the odds. Boy did I underestimate the increase, though, I thought it would be two to three times more probable. It was nearly five times more probable! I think I am going to start paying closer attention to what my doctor does now. I found this subject fascinating, a discovery in an area I never thought about and was blind to. Is the change from 0.95 to 0.9025 a lowering of sensitivity, and the change from 0.05 to 0.025 a change in specificity? Or do I have that backward?

Travis - “Is the change from 0.95 to 0.9025 a lowering of sensitivity, and the change from 0.05 to 0.025 a change in specificity?”
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Sensitivity:
You nailed the sensitivity. Let's refer to our double test as Tom's test. Tom's test is just the combination of Jack and Sue's test. Tom's test is positive when Jack and Sue's is positive. Tom's test is negative when either (or both) Jack and Sue's tests are negative.

Sensitivity of Jack's test is P(J|F), probability of Jack's test being positive given someone has the floobieitis, or 0.95

Sensitivity of Tom's test is P(JS|F), probability of Jack and Sue's tests being positive given someone has the floobieitis, or 0.95 * 0.95 = 0.9025

You will notice that counter intuitively, Tom's test is less sensitive than Jack's test!
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Specificity:
You kind of messed up specificity a little. First off 0.05 * 0.05 = 0.0025, but more importantly this not the specificity, but it is related to the specificity. What you described is the false positive rate. The specificity is the complement of the false positive rate. The complement of any quantity is 1 minus that quantity. So:

False positive rate of Jack's test = 0.05
Specificity of Jack's test is 1 – 0.05 = 0.95

False positive rate of Tom's test = 0.05 *0.05 = 0.0025
Specificity of Tom's test = 1 – 0.0025 = 0.9975
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Jack's test is better at finding potential people with the disease, but it generates a lot of false positives. Tom's test doesn't find quite as many people with the disease, but it generates far fewer false positives. Which test is better? Well that kind of depends on your goals.

Also a side note before anyone goes and tells their doctor they don't want to take a test because they aren't as reliable as we assumed they were in the past:

Remember all of these calculations were for a situation where you are just randomly testing members of the population, and 1% of the population has the disease. If you were to go to the doctor with the symptoms of sweating mercury, your hair spontaneously lighting on fire, and one of your legs has fallen off---all symptoms of floobieitis---the probability you have the disease is likely much higher than 1%. Remember all of the calculations had these terms:

P{F} = probability someone in the population has the disease, has not changed = 0.01
P{~F} = probability someone in the population does not have the disease, has not changed = 0.99

Let's say the probability of someone having floobieitis taken randomly from a population of people WITH THAT LIST OF SYMPTOMS is 60% then our terms will look like:

P{F} = 0.6
P{~F} = 0.4

Then after getting a positive result from Jack's test, the likelihood you actually have floobieitis would be about 97%!

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