Happy New Year!
This month’s challenge is based around statistics, specifically permutations and combinations around shuffling a deck of cards. It’s inspired by this article:
Your challenge, should you choose to accept it, is to perform the following in Mathcad Prime:
- (Basic) Calculate the number of ways to shuffle a deck of cards. (Standard 52 card deck, no Jokers).
- (Basic) Calculate the number of ways to shuffle a deck of cards if the suit (heart, diamond, spade, club) doesn’t matter.
- Calculate the number of ways to shuffle a deck of cards consisting of just the face cards (jacks, queens, and kings). If you could shuffle the deck one time a second, how long would it take to shuffle each possible way? (Express the result in hours, days, or years, whichever is easiest for people to grasp.)
- How long would it take to shuffle the face cards until there’s a 0.5 probability you’ve shuffled the same way twice? If you add the aces, how long would it take for that same 0.5 probability that you’ve shuffled the same way twice?
- Create a Chart Component to depict the number of ways to shuffle a deck from 1 to 12 cards. (Hint: use the Chart Component’s logarithmic scale capability.)
- (Optional) How many cards would be in a deck in which the number of ways to shuffle would be on an order of magnitude with the number of atoms in the Earth? What about the universe? Assume 10 to the 50th power for number of atoms in the Earth and 10 to the 80th for the universe.
- (Optional) Add a header and footer to your worksheet.
As always, documentation is key. Create your worksheet in a manner that someone can understand it on its own.
Find the Mathcad Community Challenge Guidelines here!
OK, I'll jump in as its still January 1st over here and my avatar feels addressed by the task 😉
Some remarks:
I wouldn't say that the task is based around statistics but rather enumerating combinatorics and basic probability.
I don't think you really mean the number of ways we can shuffle a deck of cards. Because this would mean that we have to count methods like Overhand shuffle, Riffle (Dovetail) shuffle, Hindu shuffle, etc, -> https://en.wikipedia.org/wiki/Shuffling
You rather sure mean the number of ways the cards can be arranged in sequence.
When you ask for the 50% probability you say "How long would it take to shuffle the face cards until there’s a 0.5 probability you’ve shuffled the same way twice? "
That's a bit unclear and you sure mean a probability of 50% or higher, don't you? Otherwise I guess there would be no solution.
Furthermore I guess you may mean twice or more and you would also be happy if the card arrangements achieved contain even more than just one pair of duplicates, correct?
I interpret that question similar to the well known Birthday-Problem, where the question is about the probability of at least two persons having the same day of birth (ignoring the year and Feb 29th). Just with 12! and 16! instead of 365.
Does my interpretation match your intention?
For the last but one (the first optional) I had to use the symbolics of Mathcad 15 as Primes was not capable enough and I did not want to resort to approximations and messing around with trial and error.
At least I found a way to get a solution good enough for this task even with Prime by using the Stirling formula and a little trick.
Anyway, find attached my 2 cents worth...
EDIT: Had to exchange the pdf file for one created by a different program because the one created using Primes "Save to pdf" option had some strange errors like (erroneous spaces highlighted in yellow by me)
