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!