Question 1086535
<font color="black" face="times" size="3">*[Tex \Large P(A) = 1-P(\overline{A})]


*[Tex \Large P(A) = 1-\frac{2}{3}]


*[Tex \Large P(A) = \frac{1}{3}]


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Now create a Venn diagram. I'll call this figure 1.


<img src = "https://i.imgur.com/Bsrau31.png">


The overlap is important here (as you'll see in figure 2). The fact that *[Tex \Large P(A \cap B)] is nonzero indicates that overlap is needed between sets A and B. 


Note: U is the universal set. 

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Now turn to figure 2.


<img src = "https://i.imgur.com/76bDL5U.png">


The change here is that I've added two shaded regions. The green region is the portion inside circle A but it is not in circle B. The area of this green region will be exactly equal to the probability your book wants for part (iii). In other words, 


area of green region = *[Tex \Large P(A \cap {\overline{B}})]


Note how the orange region is the remaining part inside circle A, and its also in circle B as well. The orange region is the area corresponding to the probability *[Tex \Large P(A \cap B)]. So this means,


area of orange region = *[Tex \Large P(A \cap B)]


The green and orange regions do not overlap. The regions combine to compose set A. They completely <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">partition</a> set A. 
This is an example of <a href="https://en.wikipedia.org/wiki/Law_of_total_probability">The Law of Total Probability</a>


Because of this fact, we can say


*[Tex \Large P(A \cap {\overline{B}}) + P(A \cap B) = P(A)]


*[Tex \Large P(A \cap {\overline{B}}) + \frac{1}{4} = \frac{1}{3}]


*[Tex \Large P(A \cap {\overline{B}}) + \frac{1}{4} - \frac{1}{4} = \frac{1}{3} - \frac{1}{4}]


*[Tex \Large P(A \cap {\overline{B}}) = \frac{1}{3} - \frac{1}{4}]


*[Tex \Large P(A \cap {\overline{B}}) = \frac{4}{12} - \frac{3}{12}]


*[Tex \Large P(A \cap {\overline{B}}) = \frac{4-3}{12}]


*[Tex \Large P(A \cap {\overline{B}}) = \frac{1}{12}]


Answer: <font color=red>1/12</font> (which is approximately <font color=red>0.0833</font> in decimal form)</font>