In many movies, some people remember the cards that are played in poker games. Is this possible to remember all the cards to predict next 10 or 20 hands? If yes, how and is it allowed to do that in casino's?

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    A single poker deck is shuffled each hand. If you dealt out of a shoe with multiple decks then you could get 5 of a kind. It would break the game.
    – paparazzo
    May 2 '16 at 17:05

In casino poker games, the cards are reshuffled for each hand, so "counting" would be useless. Shuffle tracking might be possible, but the edge from doing so would be dubious. A casino would be wasting its time to look for or prevent such things.

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