The Humanly Infinite

today I want to return to the topic of large numbers specifically my favorite large number 52 factorial when I originally discussed this number three repeating comments kept popping up first roughly 10% of comments joked about the idea of 53 factorial so congrats on the originality secondly many didn't like that I said it was equal to the concept of infinity so I want to return to the idea of humanly infinite or infinite as far as humans are concerned but the most interesting idea was well of course the odds of your deck of cards having existed is zero but surely two decks have had the same order at least once in all of history and this was interesting because it certainly seems possible the birthday Paradox or problem explains the unintuitive reality that if you have 23 people in a room there is a 50% chance that two people in that room share a birthday the reason this number is so low is because we're not simply finding out if one person shares a birthday with anyone else but if anyone shares a birthday with anyone else if you know anything about probabilities and statistics you likely know about independent events like flipping a coin if I ask you what's the probability of flipping three heads in a row then that is simply 1/ 12 to the power of three but if I ask you if you flip a coin three times what's the probability you get heads twice in a row well that is no longer a singular independent event you could get two heads in a row in three different ways therefore to calculate the probability of this event you can either multiply the probability of one of these events times the number of ways it's possible or you can divide the number of ways it can occur by the total number of possibilities this logic applies to the birthday problem pretend there was a person in a room another person enters there are now 365 squared possible birth combinations of which 365 share a birthday when a third person enters the room now there are 365 cubed combinations I'm using 50 dots for the side lengths now because 365 cubed is 48 million dots but the pattern is the same now there are about 399,000 combinations that share a birthday we divide we get 8210 of a per. we could continue this with the fourth person but it's very tedious the easier method is to calculate the probability that everyone doesn't share a birthday I won't explain this method as it has been done before and I wouldn't be able to add anything new to the discussion I recommend this video if you're curious but if we take this route we can use a handy dandy equation to Simply plug in values if we were to plot the probability of two people sharing a birthday given the number of people we would find that at 23 people that probability crosses the 50% threshold this occurs because the more people you add the more pairs or opportunities you create for two birthdays to be shared therefore we get a match sooner than we'd expect what about our playing card example for playing cards this same logic applies except instead of 365 birthdays we have 52 factorial Unique Card combinations we can reuse this equation except this factorial really just messes everything up imagine 52 factorial factorial being a part of the calculation this is wild thankfully we can just rewrite these equation as so now we can do some fast calculations first what if we were to flip a coin or how many decks of cards do we need to shuffle until there is a 50% chance two decks were the same well that number is roughly 10 to the 34th power that isn't absolutely insane number we have certainly not reached that number of shuffles and we will never ever come even remotely close so I can confidently sit here and tell you of all the properly shuffled decks of cards throughout human history there is a significantly less than 50% chance two of them have been the same but can we estimate what is the current probability that two decks of cards were properly shuffled into the exact same order I'm going to play it safe and make a widely conservative estimate a very rough internet search says modern poker cards were created in 1480 let's say January 1st 1480 the French distributed billions of decks of cards around the world and every day since a billion decks are properly shuffled this is of course a wild overestimation but when we are dealing with the magnitude of 52 factorial it doesn't make a difference that means today as of writing this script there have been 198 trillion deaths deck shuffles we can then plug that into our birthday Problem Solver to see that equates to a probability of 2.4 * 10 -40 or 1 and 4 * 10 39 wow honestly I thought these odds would have been like one in a thousand or one in a million but this is what does this number even mean well I went to the atomic scale I figured out what volume consists of 4 * 10 39th water molecules turns out this is a volume of 122 cubic kilm this is slightly more than Lago viedma in Argentina the odds of two people picking a single water molecule out of this Lake and those two water molecules being the same are about the same likelihood that two decks of properly shuffled cards have been the same which as far as I'm concerned is zero it's not happening now let's return to my claim that it is not just freakingly insanely big but humanly infinite by that I mean there is no non-abstract way to fit 52 factorial into a human or Humanity's world but there is one way I wanted to explore electricity at first I wanted to approach this digitally I was thinking maybe computations performed by Humanity CPUs is now run at gigahertz or billions of times per second perhaps that might get close but no before even doing the math I realized I wasn't being ambitious enough starting at a billion is too low but what about the number of electrons we've used as Humanity does this come close to 52 factorial I'm writing this part of the script before I do the calculation because I don't want to spoil my excitement but I feel I have drastically underestimated 52 factorial again if we can correct 40 as an exponent I will be satisfied I worry that we will fall shockingly short to tackle this I need to find global electricity production for the last 100 years or so most data sets only go back to the 80s however I found this graph plotting electricity production from 1900 through 2017 I then fit a curve and mesh to the graph and blender to calculate the area under the curve I then added this to the electricity produced through 2022 from another Source this came out to 829 Patt hours that's great and all but the total electrical output doesn't tell me much about the number of electrons I need to know the voltage of this electricity as higher voltage requires fewer electrons produce the same amount of energy I found this map of voltage by country and since we are dealing with such large numbers I don't need to be super accurate here so I'll just average between these two with a slight weight to the blue with this information I can make make a rough calculation of the number of electrons we have used since 1900 first we need to convert our Patt hours to Jews which is dishearteningly low now that we know how much energy we need to produce we need to know what combination of charge and voltage will create that energy voltage is electrical pressure or the ferocity at which a charged object wants to move if we picture an electron on a Ledge falling through a wheel a higher voltage means the electron will fall with greater kinetic energy spinning the wheel more than electron who fall slower at lower voltages it should be clear why we need less electrons produce the same amount of energy if the voltage is high well I've decided our voltage will be 190 volts the energy we can extract from our voltage field is simply the charge of the particle in KMS times the voltage but we don't know the charge so we'll have to solve for that doing so gives us a charge of 15 million trillion KS now we simply have to divide this value by the charge of an electron to see how many electrons it takes to produce this charge 9.8 * 10 the 37th electrons so close to that 40 exponent well I guess not really it's three orders of magnitude off but you know three numbers feels close to me also I want to point out this number is still less than the previous number so you're more likely matching up two random electrons from a set of all electrons Humanity has ever used than two decks of cards having ever Shar the same order in human history 52 factorial is humanly infinite there's nothing we could do to produce 52 factorial as a real Concept in the world over a human's lifespan but we've come so far can it be done let's say a person lives 100 years moving backwards how much energy needs to be distributed by Humanity over those 100 years to produce 52 factorial electrons that many electrons equals a charge of 1.3 * 10 49th kons at 190 volts this would equal 2.45 * 10 51st Jews 100 years is 3.15 billion seconds if we divide our total energy by that many seconds then Humanity would need to Output 7.76 * 10 41 Jew every second which is about 100,000 times more energy than the entire Milky Way produces so no 52 factorial will forever be Out Of Reach For [Music] Humanity

Channel: But Why?

Views: 238,990

Keywords: birthday problem, birthday paradox, 52!

Id: RdnVhjYFr7w

Length: 10min 55sec (655 seconds)

Published: Wed Dec 20 2023