I was reading the book I came across the conditional probability. The statement is:
Imagine a certain disease that affects 1 in every 10,000 people. And imagine that there is a test for this disease that gives the correct result (“diseased” if you have the disease, “nondiseased” if you don’t) 99% of the time.
T = Test result is positive. D = Person is diseased.
So, P(T|D) = 0.99 and P(D) = 0.0001. The author calculated P(D|T) by Bayes theorem.
P(D|T) = P(T|D) P(D) / [P(T|D)P(D) + P(T|~D)P(~D)]
For that, he calculated different probabilities.i.e.
- P(~D)= 1- P(D) = 0.9999
- P(T|~D) = 1 - P(T|D) = 0.01
I could understand intuitively that
P(~T|D) = 1 - P(T|D).
But I am unable to understand how
P(T|~D) = 1 - P(T|D) is valid?
Can somebody explain mathematically?