I kept thinking about the probability posts earlier in this group. We were wondering if there was a way to increase the probability that just by looking at coincidences between two really random decks, you would obtain at least one card. The idea was to have a process of selection different from "think of any card/pick any card" so that the spectator could wonder whether or not you indeed handled all the shuffle/cut matters, etc.. But there was the problem that occured in 35% of the trials, in which you and your spectator spent one or two minutes looking at 2 x 52 cards to finally find out no card had been chosen...

Well, I've found something that may interest you. It is at the same time "really random" but :

a) it produces a result more than 98% of the time (for a 32 card deck as well as a 52 cards one).

b) it produces this result in less than 15 cards 85% of the time (for a 32 cards deck), or in less than 22 cards 85% of the time (for a 52 cards deck).

c) it has a subtle bias so that you can make a "general" prediction on the chosen card that will be right 85% of the time (with a 52 cards decks), or 75% (with a 32 cards).

The method :

The spectator shuffles his deck.

He deals the cards face up one by one, and he counts at the same time "Ace-2-3-4-5-6...-King-Ace-2-.." or "Ace-7-8-9-10-Jack-Queen-King-Ace-7..."

When he has found the card "at the right place", it's over.

The "general" prediction is : "It's not a face card".

The math-inclined readers here have already realized that by just looking at the value, not the colour, as far as probability of success is concerned, it's as if you were making the "two decks-coincidence choice" four times in a row. Hence the a) result.

The even more math-inclined readers here will also recognize a method a bit like the one behind Poisson's law : hence the b) result taken from (1-exp(-n/13)) and
(1-exp(-n/8)) formulas.

Finally, one can show that the probability of having the first success at Nth card decreases with N. Since you're putting the face cards at the end of the cycle in the test, you obtain the c) result... Contrary to the two decks-coincidence choice in which you had no way to bias the result.

Computations show that while, in a 32 cards deck, face cards are 3/8 = 37.5% of the cards, with this method, you obtain them only 25% of the time. In a 52 cards deck, face cards are 3/13 = 23.1% of the cards, but this method give them only 15% of the time. In this latter case, I think it's worth insisting with a smile "don't worry if you don't pick a face card" before the test.

[For the info, with a 32 card deck, if the spectator counts "7-8-9-..-King-Ace-7-8...", the probability of having a face card goes from 25% to 28%, since the 3 face cards are one rank earlier in the cycle].

So take your 52 card deck, shuffle it, deal it as I told you, and don't worry if the one at the the correct place is not a face card