Math

Last week I discussed the probabilities of three effects: ACAAN, Triumph, and sympathetic Rubik’s cube. The Out of this World effect slipped my mind last week so I’ll mention it today.

We’ll use magicpedia for the definition of the effect.

A spectator separates a deck of cards into reds and blacks without looking at the faces of the cards.

Unlike facing, card color is not independent. There are only 52 cards in a deck so if you know the color of the first 51, you can determine the color of the last card with total accuracy.

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