shuffle

imagine a deck with 3 cards (so elements 0, 1 and 2 of an array):

shuffling the bad way..

for each of the original cards[0] to cards[2]:
  choose a random card between cards[0] and cards[2]
  swap the original card with the random card

• there is 3 possible combinations here:
• 123, 132, 213, 231, 312, 321
• or in other words "3!" (which means 3 factorial = 3 * 2 * 1 = 6)
• so if we shuffled the deck 600k times, we should see each combination around 100k times
• but we dont with this algorithm..:

• because of "for each card, choose a random card between cards[0] and cards[2]"
• this implies 3³ combinations = 27 outcomes
• in our heads though, we know this isnt possible.. there are only 6 (3!)
• more combinations then real world outcomes = some combinations being over-represented
• which means bad shuffling!
• more shuffling makes results here even worse (meaning the over represented probabilities become even more obvious)
• more cards makes result worse too.. e.g. a 6 card deck = 6⁶ combinations (46,656) trying to fit into 6! = 720 real world combinations

shuffling the better way..

for each of the original cards[2] counting down to cards[0]:
  choose a random card between cards[0] and the original card im currently at
  swap the original card with the random card

• starts with swapping from the third position with any of the three cards
• so it can stay in 3rd (123), or swap places with 2nd (132) or 1st (321)
• then 2nd can stay in 2nd or swap with 1st..
• so depending on what 3rd did gives us either:
• (123 = 123 or 213), (132 = 132 or 312) and (321 = 321 or 231)
• if you work it out, thats 6 possible combinations!
• each with almost perfectly equal chance.. so good shuffling

shuffle: java

import java.util.Random;
public class shuffle{
  public static void main(String[] args) {
    //*****SET UP 2 DECKS*****:
    int[] cardDeck = new int[52];
    int[] tempDeck = new int[52];
    for(int count=0; count<52; count++) {
      cardDeck[count] = count;
      tempDeck[count] = count;
    }
    
    //*****NOW SHUFFLE*****:
    Random seed = new Random(); 
    for(int number=52; number>=1; number--) {
      int numbersLeft = seed.nextInt(number);
      cardDeck[number-1] = tempDeck[numbersLeft];
      tempDeck[numbersLeft] = tempDeck[number-1];
    }
    
    //*****CHECK RESUT*****:
    for(int test=0; test<52; test++){
      System.out.println(cardDeck[test]);
    }
  }
}