This web page is an amazing tutorial for application of Bayes’ theorem. One of the problem resolved by this theorem is:
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
This answer is given here: 
And the formula deduced from Bayes’ Theorem giving P(positive|Cancer) is: 
An Intuitive Explanation of Bayesian Reasoning
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