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