From 'Blackjack' To The Universe (Part 2)
Welcome back travellers! I hope you found the first installment as mind blowing as I did. (I'm not trying to big-up my ego there, I'm not fussed if you enjoyed the way I write, it's the content that completely blows me away!)
Just to recap, I am trying to show why it is highly unlikely that any two randomly shuffled decks of 52 cards have ever been the same, even assuming humans could play with cards since the Big Bang! Wow! What a statement! In the last post, I showed how it's possible to count the number of ways of arranging n objects (this is called the number of 'permutations' of n objects). Without going into too much detail, it works out that for n objects, there are n! permutations (where n! = n x (n-1) x (n-2) x (n-3) x ... x 3 x 2 x 1). This means that for a deck of 52 cards, there are 52!, or 8.07 x 1067 permutations. This is a massive number, and is about equal to a tenth the number of atoms in the whole Milky Way! (This was my own 'calculation', I'll pop another post up soon showing all my workings out so you can see for yourselves).
How long would it take to shuffle this many permutations? If a person had mysteriously and magnificently exploded into existence at the precise moment of the Big Bang, and had shuffled a deck of cards every second since, until this moment, they would have dealt 43 x 1015 hands. If the current population of the Earth (around 7 billion, or 7,000,000,000 ± 60,000,000) had each shuffled and dealt one hand every second ("Every second I'm shuffling"), they would have managed to shuffle their way through 3 x 1025 permutations. While 3 x 1025 is itself a very big number (3 followed by 25 digits), it is still around a million-trillion-trillion-trillion times smaller than 52!.
What is the probability that two of these 3 x 1025 permutations are identical? How do we calculate it, and what does it mean anyway?
Next week's installment will provide a summary of what we know so far and I will try to calculate the final probability of two shuffled decks of cards being identical after 3 x 1025 shuffles, hopefully showing why it is unlikely that two randomly shuffled decks of cards have ever been identical.
Just to recap, I am trying to show why it is highly unlikely that any two randomly shuffled decks of 52 cards have ever been the same, even assuming humans could play with cards since the Big Bang! Wow! What a statement! In the last post, I showed how it's possible to count the number of ways of arranging n objects (this is called the number of 'permutations' of n objects). Without going into too much detail, it works out that for n objects, there are n! permutations (where n! = n x (n-1) x (n-2) x (n-3) x ... x 3 x 2 x 1). This means that for a deck of 52 cards, there are 52!, or 8.07 x 1067 permutations. This is a massive number, and is about equal to a tenth the number of atoms in the whole Milky Way! (This was my own 'calculation', I'll pop another post up soon showing all my workings out so you can see for yourselves).
How long would it take to shuffle this many permutations? If a person had mysteriously and magnificently exploded into existence at the precise moment of the Big Bang, and had shuffled a deck of cards every second since, until this moment, they would have dealt 43 x 1015 hands. If the current population of the Earth (around 7 billion, or 7,000,000,000 ± 60,000,000) had each shuffled and dealt one hand every second ("Every second I'm shuffling"), they would have managed to shuffle their way through 3 x 1025 permutations. While 3 x 1025 is itself a very big number (3 followed by 25 digits), it is still around a million-trillion-trillion-trillion times smaller than 52!.
What is the probability that two of these 3 x 1025 permutations are identical? How do we calculate it, and what does it mean anyway?
Next week's installment will provide a summary of what we know so far and I will try to calculate the final probability of two shuffled decks of cards being identical after 3 x 1025 shuffles, hopefully showing why it is unlikely that two randomly shuffled decks of cards have ever been identical.
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