Example (Using Bayes’ without definition):
Two boxes:
The first contains two green and seven red balls. The second contains four green balls and three red balls.
Bob selects a ball by
1.) First choosing one of the two boxes at random. 2.) Selecting one of the balls in this box at random.
What is the probability that he selected a ball from the first box, given that the ball is red?
find p(box1 and red) and p(box 2 and red)
1.) 2.)
Therefore p(red) =
Bayes’ Theorem Formula
Note: the line above f means complement of; 1-p(F)
Example using this new idea:
Suppose 1 out of every 1000 computer chips has a defect. The test to discover defects is successful with a probability of .99.
If the chip does not have a defect, the test will inaccurately report that it does 0.02 probability.
Q: If the outcome of a test indicates that there is a defect, what is the likelihood that the chip is actually faulty.
p(actually faulty given the test says there is a defect)
so actually faulty is F, test says defect is E
(probability there is defect given the test is faulty) * (probability actually faulty)
/
probability (defect given faulty)(actually faulty + (Defect but not faulty)(not actually faulty))
ok break down these statements one more time:
(defect given faulty): ‘
If the chip has a defect, the test will discover the defect with
probability 0.99.’
(actually faulty): ‘0.001 because one out of every 1000 chips has a defect.’
(Defect but not actually faulty): ‘If a chip does not have a defect, the test reports that
it has a defect with probability 0.02.’
(not actually defect): .999 (1 out of 1000 are defect so 1-.001)
plugging these values into formula gives 0.047 ^^
Apparently this has applications in spam filters lol.
this note was imported from my other vault, ideas are not complete :(