A draws two cards at random from a pack of $$52$$  cards. After returning them to the pack and shuffling it, $$B$$ draws two cards at random. The probability that their draws contain exactly one common card is :

Solution :
The probability of both drawing the common card $$x$$
$$ = P\left( x \right) = $$   (probability of $$A$$ drawing the card $$x$$ and any other card $$y$$) × (probability of $$B$$ drawing the card $$x$$ and a card other than $$y$$)
$$\therefore \,P\left( x \right) = \frac{{{}^{51}{C_1}}}{{{}^{52}{C_2}}} \times \frac{{{}^{50}{C_1}}}{{{}^{52}{C_2}}}$$     for all $$x,$$ where $$x$$ has $$52$$  values.
$$\therefore $$  the required probability $$ = \sum {P\left( x \right)} = 52 \times \frac{{51 \times 50 \times 4}}{{52 \times 51 \times 52 \times 51}} = \frac{{50}}{{663}}.$$