Study questions regarding shuffling a deck of cards:

In order to tackle these study questions, be sure you have read the brief explanation of a mathematical model for shuffling a deck of cards. If you have not read this page, click here.

The study questions also correspond to a Java applet simulating this model. To open the applet in its own window, click here.

1. What would have to happen for only 1 rising sequence to occur after the deck is shuffled once? Try running the experiment a large number of times, say 1000 times. Does this happen? How probable is it?
2. In order to determine the number of rising sequences after a given shuffle, a one-to-one correspondence must be chosen between the set containing the 52 playing cards and the set {1, 2, 3, ..., 52}. Otherwise, how do you know if the 2 of hearts is greater than the 3 of clubs? Use the applet to determine the exact one-to-one correspondence chosen for the simulation. Determine a few other choices that could be made.
3. Is it possible to have 3 rising sequences after just one shuffle? Why or why not?
4. If the average number of rising sequences is R_avg then the average size of a rising sequence is 52/R_avg. Do you expect most rising sequences to be close in size to 52/R_avg or do you expect some small rising sequences and some large rising sequences? Experiment with the "color" option in the applet to see if the reality meets your expectations.
5. Press the reset button a number of times. From the top panel, notice the average number of rising sequences after successive shuffles. Looking at the bar charts, determine at which shuffle it becomes difficult to predict whether or not the number of rising sequences will rise or fall? For instance, if the deck has been shuffled 3 times, do you expect the number of rising sequences to rise or fall on the next shuffle? What about after 6, 7 or 8 shuffles? This is an important aspect in the determination of the randomization of the deck! You are determining the point when continued shuffling no longer helps make the deck less predictable!
6. Enter the number 500 into the textbox. Press the "Run simulation" button. From the bar charts in the top panel, what is the average number of rising sequences after 7, 8 or 9 shuffles? Now, click the bar corresponding to 7 shuffles, does the histogram from your experiment depicted in the middle panel correspond to the probability of arriving at each number of rising sequences? Is the same true for 8 and 9 shuffles?
7. Now enter the number 1 into the textbox and click the "Reset" button. Click the reset button a number of times. When the number of rising sequences is greater than or equal to 29, does the next shuffle tend to produce more or less rising sequences on the next shuffle?
8. Which do you think is more likely after a large number of shuffles, to return to a nearly ordered deck (with very few rising sequences) or to return to a very unordered deck (with numerous rising sequences)? Test your conjecture by typing the number 1,000 in the textbox and pressing the "Run Simulation" button. The middle panel of the applet for shuffles 3 through 9 should assist in whether you believe your conjecture!

These web pages have introduced and visualized the card shuffling mathematical model developed by D. Bayer and P. Diaconis. (Ann. Appl. Probability 2, 294-313, 1992) Their model demonstrated that 7 riffle shuffles of a deck of cards produces acceptable randomization. This result was made popular by G. Kolata in his article in the New York Times. (sec. C, p.1, Jan 9, 1990)

Bayer and Diaconis justify their claim through the use of a staticical measure known as the "total variational norm." A description of this norm is beyond the scope of these pages. However, your insights in the above study questions should help you see that the number of rising sequences indeed stabilizes after 7 shuffles. This implies that after 7 shuffles continued shuffling no longer helps make the deck less predictable.

Yet, this all depends on your definition of randomness! To read more about card shuffling models, you are encouraged to read the following:

• MathTrek: Disorder in the Deck from Science News Online (October 14, 2000)
• Rising Sequences in Card Shuffling from Mudd Math Fun Facts produced by Harvey Mudd College.