バイアスシャッフル - 最後編

Continuation of this post

For the final part, we discuss the simplest, yet most effective method of randomizing cards relative to their initial position: Pile shuffling.

Pile shuffling is when you take your deck, and deal piles. The effectiveness of randomization is based on the number of piles you create; the more piles the farther apart cards will be when their initial order.

In the interest of time, I won't bother to take pictures, mainly because pile shuffling is the only kind of shuffling discussed so far that you can perform repeated replications without any deviation (in other words, the randomness of pile shuffling is the fact that there is no randomness to begin with). Basically, you won't need visuals to interpret the result.

The best way to explain it is to once again consider our sample space:

Si=[1D, 1C, 1H, 1S, 2D, 2C, ... 10S]

If we did pile shuffling, making 4 piles of 10, and then stacked each pile on top of one another, then we would get:

Sf=[10D, 9D, 8D, 7D, ... 4S, 3S, 2S, 1S]

I am not listing the entire series, as it can easily be derived from the given information.

What happened here? Originally, the cards were ordered from lowest number to highest number, lowest suit to highest suit. After performing the controlled experiment, we find two interesting results:

1) All the suits are together
2) The numbers go in descending order, as opposed to ascending of the original

Both results do one thing in common: They separate all cards from its original position. 

Take for example, 3C. Originally, 3C is relative to 3D and 3H. After the experiment, 3C is now relative to 4C and 2C. Picking any other card from our sample space and we'll notice the same thing: there is a deviation of 4 cards from the original.

Note that this is only the case for all cards because we divided the deck into 4 piles, which is perfectly divisible in a 40 deck. Similarly, if we did 2 piles, there would be 2 deviations, and if we did 8 piles, there would be 8 deviations. If we did a non-divisible number, such as 3, then most cards would be 3 deviations, where the others would exhibit 2 or 4.

Pile shuffling is basically stacking, since you control where each card is relative to another card. Granted, most people don't have an analytical mind that can take full advantage of this, but it's a nice fact to know. The point of pile shuffling for the layperson is to separate their cards close to each other, and then use the other two methods of shuffling (standard and cascading) to randomize the newly separated deck.

Practically, you would only pile shuffle after a game, whereas if you had to search your deck for a card, then you would standard and cascade, in the interest of time.