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This might be a super novice question you'll all laugh about, but you have to be brave if you want to seek knowledge! So here we go:

Let's say there's a medical test that's 99% accurate.
Let's say 100 people go through this test (and we know their true state).

Using the Binomial distribution we can calculate that:

  • P(test results were accurate for 100 of the 100 people): 0.366
  • P(test results were accurate for 99 of the 100 people): 0.3697
  • P(test results were accurate for 98 of the 100 people): 0.1848
  • P(test results were accurate for 97 of the 100 people): 0.06

Here's my philosophical question:

Why is the probability that the results were accurate 100/100 is about twice the probability the results were accurate for 98 of 100 subjects?

Intuition behind the question: in both cases - being accurate 100/100 of the time and 98/100 of the time, the test was 'off' from what we'd expect it to be by one person. So, why isn't the probability the same whether we're off 1 to 'the good' or off by 1 to 'the bad'?

Followup - (assume it's related) - Why is the probability the test was correct for 97 of the 100 subject approx. a third of the probability it was accurate for 98/100 subjects?

Thanks for your inputs!

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    $\begingroup$ Given the probability of success per person, 0.99, and the number of people, 100, the relevant binomial distribution is very far from being symmetric. If the probability of success were closer to 0.5 and/or the number of people much larger, the distribution would be closer to being symmetric. You need to adjust your intuition to account for asymmetry. $\endgroup$ – Mark L. Stone Dec 13 '15 at 15:30
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    $\begingroup$ Welcome! You're right - After you give it some thought, you have to ask. Otherwise it's easy to get discouraged and never find out the answer. $\endgroup$ – Antoni Parellada Dec 13 '15 at 15:50
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Think about a similar question.

Using your setup

  • P(test results were accurate for 3 of the 3 people): 0.97
  • P(test results were accurate for 2 of the 3 people): 0.03
  • P(test results were accurate for 1 of the 3 people): <0.001

Do these results seem strange?

The test is very accurate so the probability that it would perform poorly is low. Just like in your example the three results are off by 1 person. However in this situation it is hopefully more clear why the first scenario is much more likely than the third.

"So, why isn't the probability the same whether we're off 1 to 'the good' or off by 1 to 'the bad'?" Because the test is very accurate so it is much more likely to be off in the good direction than the bad.

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