Refer to the tree diagram in Figure 5.7. Suppose you want to find the probability P(B|A) using the information available in the tree diagram. To do this, P(B|A) must be expressed in terms of conditional probabilities, like P(A|B) and P(A’|B). a. Use the addition law to show that P(A)= P(A and B)+ P(A and B’).
Refer to the tree diagram in Figure 5.7. Suppose you want to find the probability P(B|A) using the information available in the tree diagram. To do this, P(B|A) must be expressed in terms of conditional probabilities, like P(A|B) and P(A’|B).
a. Use the addition law to show that P(A)= P(A and B)+ P(A and B’).
b. Use the conditional probability formula to write P(A and B) in terms of P(A|B) and P(B). Develop a similar formula for P(A and B’) in terms of P(A|B’) and P(B’).
c. Use parts (a) and (b) to show that: formula in attched image. This formula, known as Bayes’ theorem, is used to “turn conditional probabilities around”; that is, it allows us to express P(B|A) in terms of P(A|B) and P(A|B’ ).
d. In Figure 5.7, the probability associated with any path from left to right through the tree is simply the product of the probabilities of the branches. Why?
e. Use the observation in part (d) and the conditional probability formula for P(B|A) to justify Bayes’ theorem.
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