# Probability for Magic Trick

I had a probability problem to solve , but could not proceed further , we have a m identical decks having n cards , where each deck has n different cards . Now shuffle them and select n cards . Now a someone chooses a card from the new deck , I do not know the card but I also take card after he has put it back , what would be the probability I would pick the same card. So any hint or method to solve would be appreciated.

Let $x$ denote the card drawn from the new deck. The probability that the new deck contains (exactly) $k$ copies of $x$ will be $$P(k)=\frac{\binom{m}{k}\binom{mn-m}{n-k}}{\binom{mn}{n}}$$ The probability you seek will then be $$P=\tfrac{1}{n}\cdot P(1)+\tfrac{2}{n}\cdot P(2)+...+\tfrac{m}{n}\cdot P(m)$$ Unless of course $m>n$ then you must truncate the above sum at $\frac{n}{n}\cdot P(n)$. maybe the parameter mn-n should be mn-m