How to do Bayes’ theorem?

Use Bayes’ theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places.
P(A | B) = .9, P(B) = .6, P(A | B’) = .8. Find P(B | A).

We don’t have an example like this in class so I was wondering if someone could help me figure out how to solve it.

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Here are a couple of hints: State Bayes’s Theorem. Can you calculate P(B′)P(B^\prime), which is the complement of P(B)P(B)? How about P(A)P(A)? Can you use Bayes’ theorem now?
– Larry B.
2 days ago

  

 

Is P(B’)=.4? I still don’t think I am doing it correctly
– user344249
2 days ago

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1 Answer
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Bayes theorem:

P(B|A)=P(A|B)P(B)P(A)P(B|A)=\frac{P(A|B)P(B)}{P(A)}

So you only need to find P(A)P(A). Use the law of total probability:

P(A)=P(A|B)P(B)+P(A|B′)P(B′)P(A)=P(A|B)P(B)+P(A|B’)P(B’)

and notice that P(B′)=1−P(B)P(B’)=1-P(B)

  

 

Thank you so much for explaining that all out like that, I seriously appreciate it
– user344249
2 days ago

  

 

I am glad that it helped @user344249
– msm
2 days ago