I am reading Nate Silver's book "The Signal and the Noise" and have a question about Bayes Theorem. I've created my own example and am trying to wrap my mind around the conclusion.
Let's say, before any information, I think there is a 5% chance humans have caused global warming.
Then, I hear information that scientists think there is a 99% chance that humans have caused global warming.
I also know that the probability that the 99% claim is wrong is 10%.
Using the Bayes Theorem calculation, the result is a 34% chance that humans cause global warming.
Here is the calculation:
X = initial probability of humans causing global warming = 5%
Y = probability of humans causing global warming, given scientist evidence = 99%
Z = probability of humans not causing global warming, given scientist evidence = 10%
The formula presented in the book (page 247) is:
Revised probability (given the new information) = XY / (XY + Z(1-X))
Revised probability (given the new information) = 34%
My intuition says that, after this new knowledge, the chances that humans have caused global warming is instead (10% * 1%) + (90% * 99%) or 90%.
This would be based on the fact that theres a 10% they're wrong and 90% chance they're right.
What is wrong about my application of the theorem or understanding of the theorem that causes this mental roadblock?