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"Two shuffled decks are laid out one beneath another...what is the expected number of matches" (match is same rank and suit)

Part where i am stuck-

Does the probability at each draw for a match stays same equal to $\frac{1}{52}$. Shouldnt the probability of a match on the second draw be $\frac{1}{51}$ if there was a match on the first draw

and $\frac{1}{50}$ if there wasn't a match on the first draw

How does one go from here to computing the expected value?

Edit: Looking to understand why Expected value at ith draw is always same at $\frac{1}{52}$...to me it looks like the probability at ith draw would depend on previous draws.

user377197
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    Hint: Expectation is linear. What is the expected number of matches in the $i$th position? – arkeet Oct 11 '16 at 02:26
  • I dont understand how Probability for a match at ith draw would be constant at 1/52.. – user377197 Oct 11 '16 at 02:31
  • Whatever the $i$th card in the first deck is, there is a $1/52$ chance that the $i$th card in the second deck is the same card. – arkeet Oct 11 '16 at 02:33

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