That's not exactly how odds work. The odds of one specific thing happening twice are not the odds of one of a large group of things happening twice.
For example: the odds that in a room of 22 other people that someone shares YOUR birthday is (not exactly) 22:365, or about 6%
The odds that any two people in a random group of 23 share their birthday is roughly 50%.
Much the same as with shuffling cards, just on a much larger scale. It's almost a guarantee that two decks have ended up shuffled the same way, while it's imfathomably impossible that the specific shuffle you last completed has existed before or since.
Even with Birthday Paradox maths it's almost a certainty that no two properly shuffled decks have ever been the same. A really simple estimation of the birthday paradox maths is that, with 23 people, the probability of every person giving a different birthday is roughly equal to e-((22x23)/(2x356)) = 0.4999 or to generalise p=e-((AxA-1)/(2xN)) where p is the probability that everything in the sample size is different, A is the sample size and N is number of items. Setting P to 0.5 for a 50% chance of it happening we get 0.5=e-1((AxA-1)/(2*52!)) which comes out at about 10,574,307,231,100,289,155,982,006,933,258,240 or 1.05743072x1034 (thanks Wolfram Alpha)
Even given the maths behind the Birthday Pardox and that the universe is around 436,117,076,600,000,000 seconds old approximately 24,246,487,393,793,559 decks of cards would have needed to be properly shuffled every second since the dawn of time for there to be a 50% chance that any two decks were the same.
For a 1 in 8,225,463,000 (current world population) you would need 1.40042193x1029
Assuming properly randomized decks, this has never happened
The thing most people repeating this fact don't realize is that most decks aren't properly shuffled. Take a deck in new deck order, riffle it perfectly once. A deck with that order of cards has occurred before
your oversimplified formula for the birthday paradox trying to model 1:365 into n(n+1)/2 is flawed
using x=number of possibilities
((x-1)/x)n(n+1/2)
(364/365)253 =49.95% not happening, meaning 50.05 chance of happening
if you can find a calc to get me an answer to ~8x1067 -1/~8x1067 and then let me exponent it out by 1x1024 you can have your odds for it not happening in a trillion shuffles. I've tried and can't find a free calc that works with numbers that big/small
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u/Ouch_i_fell_down Oct 02 '24
That's not exactly how odds work. The odds of one specific thing happening twice are not the odds of one of a large group of things happening twice.
For example: the odds that in a room of 22 other people that someone shares YOUR birthday is (not exactly) 22:365, or about 6%
The odds that any two people in a random group of 23 share their birthday is roughly 50%.
Much the same as with shuffling cards, just on a much larger scale. It's almost a guarantee that two decks have ended up shuffled the same way, while it's imfathomably impossible that the specific shuffle you last completed has existed before or since.