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
\[\Pr(C\text{ at index }1) + \Pr(C\text{ at index }2) + ... + \Pr(C\text{ at index }52) = 1\]Notice that there is nothing special about any of the \(52\) positions in a randomly shuffled deck. The chance that the card is at index \(1\) is equal to the chance of the card being at index \(2\), which is also equal to the chance of it being at index \(3\) and so on. We can write
\[ \Pr(C\text{ at index }1) = \Pr(C\text{ at index }2) = ... = \Pr(C\text{ at index }52) \]Now for any chosen \(i\) we can replace all the individual subprobabilities in the first equation with \(\Pr(C\text{ at index }i)\) and simplify.
\[\underbrace{Pr(C\text{ at index }i) + Pr(C\text{ at index }i) + ... + Pr(C\text{ at index }i)}_\text{52 times} = 1\\ 52 Pr(C\text{ at index }i) = 1\\ Pr(C\text{ at index }i) = \frac{1}{52} \]\(\frac{1}{52}\) is actually quite large in the grand scale of math. Triumph actually has a significantly lower probability of happening by chance. I’m using a mathematically idealized Triumph where the shuffle gives each card has an equal chance of ending up at any position and any facing rather than the physical Triumph where the cards are only shuffled face-down into face-up once by a riffle. The chance of the Triumph effect, that is, the selected card ends face up and all other cards are face down, happening is
\[\frac{1}{2^{52}} \approx 2.2 \times 10^{-16}\]That probability is, however, still huge compared to the sympathetic cube effect. The currently most commercial version of the effect is Venom Cube by Henry Harrius. The number of possible states for a Rubik’s Cube is \(43,252,003,274,489,856,000\). I won’t derive that number but there are explanations all over the internet, for example on the wikipedia page. The chance of a randomly mixed Rubik’s Cube matching a given cube is
\[\frac{1}{43,252,003,274,489,856,000} \approx 2.3*10^{-20}\]which is approximately 10000 times lower than the Triumph.
Surprisingly yet, the number of states on a Rubik’s Cube is much, much smaller than the number of permutations of a shuffled deck of cards. That number is
\[52! = 52 \times 51 \times ... \times 2 \times 1 \approx 8 \times 10^{67}\]Proving this and using it in a magic trick are both left as excercises for the reader.