Bayes Rule

Question

At the Campus Coffee Nook, 35% of the customers order regular coffee and 65% order flavored coffee. Of those who order regular coffee, 40% drink it black and 60% use sugar or cream. Of those who drink flavored coffee, 15% drink it black and 85% use sugar or cream.

 

a.) Find P(flavored coffee | cream or sugar). (Round your answer to three decimal places.)

Answer 1

It will probably be helpful to draw yourself a tree diagram for this.

 

First, draw two branches, one representing regular coffee and one representing flavored coffee. Label the regular coffee branch 0.35 and label the flavored coffee branch 0.65.

 

Now, draw two branches from the regular coffee branch, one representing black and one representing sugar or cream. Label the black branch 0.4 and the sugar or cream branch 0.6. Draw two branches from the flavored coffee branch, one representing black and one representing sugar or cream. Label the black branch 0.15 and the sugar or cream branch 0.85.

 

Multiply along the branches to find the probabilities of regular black coffee (0.35×0.4 = 0.14), regular coffee with sugar or cream (0.35×0.6 = 0.21), flavored black coffee (0.65×0.15 = 0.0975), and flavored coffee with sugar or cream (0.65×0.85 = 0.5525).

 

 

 

Now, let's use Bayes' Theorem to find P(flavored coffee|sugar or cream). I'll call it P(F|S).

 

Bayes' Theorem says P(A|B) = (P(B|A)P(A)) / P(B). So:

 

P(F|S) = (P(S|F)P(F)) / P(S)

 

 

 

P(S|F), or P(sugar or cream|flavored coffee), was given to you in the problem. If you have flavored coffee, the probability it's got sugar or cream is 0.85.

 

P(F), or P(flavored coffee), was given to you in the problem. The probability you have flavored coffee is 0.65.

 

P(S), or P(sugar or cream), can be found from the tree diagram. The two leaves with sugar or cream coffees have probabilities of 0.21 (regular with sugar or cream) and 0.5525 (flavored with sugar or cream). So, P(S) = 0.21+0.5525 = 0.7625.

 

 

 

Plug those values in:

 

P(F|S) = (P(S|F)P(F)) / P(S)

P(F|S) = ((0.85)(0.65)) / 0.7625

P(F|S) = 0.5525/0.7625

P(F|S) = 0.725