Probability question where an object is chosen randomly out of two objects I am stuck at the following probability problem. Will appreciate your help. 

Two opera singers, Mario and Clarissa both perform on the same night, in separate recitals. The independent probabilities that two newspapers X and Y publish reviews of their recitals are given below:

Probability of review in newspaper X

Mario's recital - 1/2, 
Clarissa's recital - 2/3

Probability of review in newspaper Y

Mario's recital - 1/4,
Clarissa's recital - 2/5 

Mario buys one of the newspapers at random. What is the probability that it has reviewed "both" recitals? 

I did this way: P(reviewed both recitals) = P(buys paper X)*P(X reviews Mario)*P(X reviews Clarissa) = (1/2)(1/2)(2/3) = 1/6 

But 1/6 is not the correct answer. Let me know where I am wrong and why.

Thanks in advance.
Let $B_x$ be the event "Mario buyed paper $X$", and let $B_x^c$ denote the complementary event . Let $R$ be the event "the buyed paper reviewed both recitals"

Then $P(R)= P(R \cap B_x)+P(R \cap B_x^c)=P(R\mid B_x)P(B_x)+P(R\mid B_x^c)P(B_x^c)$

Now, $P(B_x)=P(B_x^c)=\frac{1}{2}$. And $P(R\mid B_x)= \frac{1}{3} \frac{2}{3}$ ... Surely you can go on from here.
Should not it be:

\begin{align*}
P(\text{reviewed both recitals})&amp; = P(\text{buys paper $X$})\times P(\text{$X$ reviews both Mario &amp; Clarissa}) \\
&amp;\qquad+ P(\text{buys paper $Y$})\times P(\text{$Y$ reviews both Mario &amp; Clarissa}) \\
&amp;= \frac{1}{2}\cdot\frac{1}{2}\cdot\frac{2}{3} + \frac{1}{2}\cdot\frac{1}{4}\cdot\frac{2}{5} \\
&amp;= \frac{1}{6} + \frac{1}{20} = \frac{13}{60}
\end{align*}
(assuming independence of the appearance of the reviews)