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This is an example in my textbook but I do not understand it.

A news magazine publishes three columns entitled “Art” ($A$), “Books” ($B$), and “Cinema” ($C$). Reading habits of a randomly selected reader with respect to these columns are: $P(A)= 0.14$, $P(B)=0.23$, $P(C)=0.37$, $P(A\cap B)=0.08$, $P(A\cap C)=0.09$, $P(B\cap C)=0.13$, $P(A\cap B\cap C)=0.05$.

The example then goes on to say that $P(A|B\cup C)=\dfrac{P(A\cap (B\cup C))}{P(B\cup C)}=\dfrac{0.04+0.05+0.03}{0.47}=0.255$.

I do not understand how the book substituted in the numbers for $A$ and $B\cup C$. The book included a venn diagram Venn Diagram.

Could someone explain how the book got these numbers?

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When you calculate $P(B \cup C)$ you have to subtract out the intersection part after adding $P(B)$ and $P(C)$, because you would count it twice otherwise:

$$P(B \cup C) = P(B) + P(C) - P(B \cap C) = 0.23 + 0.37 - 0.13 = 0.47.$$

The numerator is the sum of the three smallest parts of $A$. $B \cup C$ is the union of the other two sets, and this intersects $A$ in those three parts.

You're given $P(A \cap B)$ and $P(A \cap C)$, as well as $P(A \cap B \cap C).$ Subtracting $P(A \cap B \cap C)$ from each of $P(A \cap B)$ and $P(A \cap C)$ gives you the probabilities you need.

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