But then suppose your friend rolls the die, quickly covers it with her hand, and takes a peek.“I can tell you this much,” she says; “it's an even number.” Now you have new information and your odds change dramatically to one in three, a 33 percent probability.One classic example of this is the presence of breast cancer among women in their forties.The chance that a woman will develop breast cancer in her forties is fortunately quite low — about 1.4 percent.
It is, rather, a statement—expressed both mathematically and philosophically—about how we learn about the universe: that we learn about it through approximation, getting closer and closer to the truth as we gather more evidence.
However, each day that he survives and the sun rises again, his confidence increases that it is a permanent feature of nature.
Gradually, through this purely statistical form of inference, the probability he assigns to his prediction that the sun will rise again tomorrow approaches (although never exactly reaches) 100 percent.
The Bayesian approach to probability is simple: take the odds of something happening, and adjust for new information.
This, of course, is most useful in the cases where you have strong prior knowledge.
But what is the probability if she has a positive mammogram?