The ACAAN effect has a \(\frac{1}{52}\) chance of happening by coincidence. Some magicians try to hype up the effect by emphasizing the fact that there are \(52\) possible cards and \(52\) possible positions and imply the probability is \(\frac{1}{52} \cdot \frac{1}{52} = \frac{1}{2704}\) but this isn’t correct.
To see why, let’s start by considering randomly chosen card \(C\) and randomly chosen position \(i\). Assuming a full, standard deck and valid card and index choices, the chance of card \(C\) being somewhere in the deck is exactly \(1\). The card must be at one of the 52 locations. That is
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