Question 1207573
<font color=black size=3>
Answer: <font color=red>0.12</font>


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 <a href="https://en.wikipedia.org/wiki/Law_of_total_probability">Law of Total Probability</a> 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') = <font color=red>0.12</font>


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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 = <font color=red>0.12</font> is the probability a student checked out a DVD but not a book.


Venn Diagram
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drawing(400,400,-5,5,-5,5,
locate(-4,4,matrix(1,5,"If","there","are","1000","students...")),
line(-4,3,4,3),
line(4,3,4,-4),
line(4,-4,-4,-4),
line(-4,-4,-4,3),

circle(-1,-0.5,2),circle(1,-0.5,2),

locate(-3.2,1.3,"Book"),
locate(2.6,1.3,"DVD"),

locate(-2,-0.5,270),
locate(0,-0.5,330),
locate(2,-0.5,120),
locate(3,-2.9,280)

)
}}}
</font>