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# Shuffle Up and Deal, and Deal, and Deal….

Take a look at the video below, where Stephen Fry, host of the British panel show QI, alleges to do something never done before:

EDIT:  Seems as though the YouTube folk removed the video clip.  Try this link instead, and let’s hope it lasts:

QI Card Shuffling Clip

What a great “hook” for a probability or counting principles unit. Some thoughts about how to use this in your class.

1.  The result given in the video can be expressed as



If we were to shuffle the cards once every second, with each arrangement occurring once, how long would it take for use to go through every possible arrangement?  A neat example of something “big”, which is accessible and easy to discuss.

2.  The online poker site PokerStars is celebrating it’s 10th anniversary, and is offering a prize to the players who participate in their 100 billionth hand (assumed to occur around the 10th anniversary).  At this rate, how long should it take PokerStars to go through all possible arrangements?

3.  As an extension, challenge your class to find the number of possible arrangements of a deck of Pinochle cards.  The main differences with a Pincohle deck are that there are only 48 cards, and each card (like the 9 of diamonds) appears twice in the deck.  This problem introduce the idea of permutations with duplicate items.  In this case, we start with 48!, but then must divide out the double-count which occur with the repeat items.  We divide by two for each instance of a repeat item, and the number of permutations is given by:



4.  Let’s evaluate Mr. Fry’s conjecture:

Were you to imagine if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people had a trillion packs of cards, and somehow they managed to shuffle them all a thousand times a second and they had been doing that since the Big Bang, they would just now begin to repeat shuffles

To summarize, we are looking at this many shuffles per second:



Dividing by the number of possible shuffles yields:



The number of seconds in each year is given by:



Which implies we would have to shuffle for this many years:



## By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

## 3 replies on “Shuffle Up and Deal, and Deal, and Deal….”

There’s a problem with Stephen Fry’s analysis. While I agree with him that it is likely that no one has ever seen that particular shuffle of the cards before, I think he has an error in his analysis of how long it would take to see a particular shuffle again. What he has calculated is the maximum number of shuffles after which we would be required to see repetition since every combination would have been seen before, not the expected number of shuffles where we see the probability of repetition rise over 50%.

I’ll have to see if I can create something to simulate this, although it will be slightly tricky as I will need more precision than most programming languages normally allow.

I agree that Mr. Fry is a bit “loose” in his analysis, confusing the number of permutations with a guarantee that all permutations would be exhausted before being repeated. An easier example occurred with the “Daily Number” here in Pennsylvania, which is a 3-digit number drawn each day. I shared with my students that my birthdate (2-14) was the last possible 3-digit number to be drawn, and it took over 7 years for 2-1-4 to occur. Certainly, many 3-digit permutations were repeated (some more than twice) during that time.

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