SOLUTION: a student goes to the library. let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) =.6, P(D)=.45, and P(D|B)=.55. Solve for P(D an

Question 1207573: a student goes to the library. let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) =.6, P(D)=.45, and P(D|B)=.55. Solve for P(D and B')
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Answer: 0.12

Explanation

Use the conditional probability formula to determine:
P(D | B) = P(D and B)/P(B)
P(D and B) = P(D | B)*P(B)
P(D and B) = 0.55*0.6
P(D and B) = 0.33

Then use the Law of Total Probability to finish things up.
P(D) = P(D and B) + P(D and B')
P(D and B') = P(D) - P(D and B)
P(D and B') = 0.45 - 0.33
P(D and B') = 0.12

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Another approach.

Consider the school having 1000 students.
P(B) = 0.6 leads to 600 students checking out a book since 0.6*1000 = 600.
Put another way 600/1000 = 0.6
Some of these 600 students also checked out a DVD.

P(D) = 0.45 means 450 students checked out a DVD (because 0.45*1000 = 450)
Some of these 450 students also checked out a book.

The notation P(D | B) is the same as saying "if we know 100% event B has occurred, what is the value of P(D)?"
Knowing that event B happened means we know the student has checked out a book.
Of the 600 students who checked out a book, 0.55*600 = 330 students also checked out a DVD.

450 students checked out a DVD
330 students did both
There will be 450-330 = 120 students who checked out a DVD but not a book.
120/1000 = 0.12 is the probability a student checked out a DVD but not a book.

Venn Diagram

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