Randomness is Random - Numberphile

Key Takeaways

I would actually see if I can test your abilities with randomness if you can do me something do me a favor so you know flipping a coin heads or tails alright so I want you to be coin flipper in your mind and I want you to flip a coin in your mind randomly 20 times and write down the result here but don't show me you've done 20 okay now I don't want to see but what's gonna happen is I'm gonna try and make money off your list so what I'm gonna do is I'm gonna give you a bunch of pennies and you don't have to pay me back they're yours and these are mine I bet to play and if I win you give me that amount and if I lose you keep that amount all right so I'm just going to put a penny down there's one penny riding on this okay so my first prediction is that you flipped ahead yes yes so buddy says head so I'm gonna take one of your coins from your stack okay I gave them to you anyway so you shouldn't be too miffed put another coin in okay I'm gonna predict tails oh so you've said head okay next I'm going to predict tails not tails next I'm going to predict tails and I'm going to bet $20 that I'm right what did you what was it heads or tails it was tails you owe me 20 bucks okay so I'm just gonna put that aside all right shall we keep playing I'm gonna bet another coin just remember you're only 20 us I wanted in us if that's okay okay I'm gonna bet another coin I'm gonna predict tails yeah it is tails okay tails it is alright so I get another coin there to predict heads I'm going to predict tails tails it is so that was another coin from you heads no okay so you get to keep that tails no it's a head so you get that heads heads it is alright so I get another coin from you this is turning out alright you remember you've got the 20 to give me okay I'm gonna predict tails yeah it is a tail yeah so I get another coin from you heads it is a head okay I'm gonna predict a tail is it a tail yeah tail okay I'm going to predict ahead okay you get that coin I'm gonna predict another head it was a head so I get a coin from you tail was a tail and I predict another tail know it's a head so I'm going to predict a head right last one 400 bucks tail yeah it is a tail that was expensive you should have listened to the way I said this you've done something fantastic for me you have been really bad at creating random sequences as I can see here the longest run that you've had is right at the beginning it was three heads and then you've pretty much alternated between tails and heads tails and heads the most run that you've had of tails or heads is two in a row and you showed a pattern there let's demonstrate something called the gamblers fallacy the gamblers fallacy is says that what's happened before has an impact of what will happen next if we were using a coin every time you flip it there's an equal probability that you've only get a head or a tail now other people think well how does that work because you know you're flipping coins surely it's not very likely to have you know five heads in a row six heads in a row but this is the nature of randomness randomness is very lumpy right and lumpiness means that you do get streaks which is part of the gamblers lexicon you know there's like we have a gambling streak so you actually have not produced a very good random sequence he didn't put enough streaks in nine this this is actually would be termed homogeneous because it's actually quite mixed the number of ways that we can have twenty heads or tails the number of permutations we can have is pretty big so we can basically get a head or a tail in each position and there's 20 positions so the total number of permutations not combinations like fixed order is 2 to the power 20 and that turns out to be 1,048,576 because you've done a permutation which has a maximum streak of either three heads or three tails you only had one little tiny streak right at the beginning which is quite interesting that is a subset of every single permutation but this is actually representative of the way you think and not just you brady a lot of people think exactly like you if they were to do random they think mixed up right that's what people think but random is it mixed up it random is what's random in actual fact it has a different nature to the idea of being mixed up because if it was evenly mixed up as I've just shown I've made a lot of money off you you end up being predictable and randomness is unpredictable so I've actually crunched the numbers and I can tell you how many permutations have a maximum of three heads or three tails as a streak 240 2830 so you can see that's a much smaller number than this and an actual fact it turns out to be 23.2% of all possible permutations it's less than 1/4 so out of all the possible permutations you can have the pool that you would choose from okay only is you're only getting one in four so can see just by that logic alone why this well it's not random all right of course I mean I wasn't consciously limiting myself those strengths of three I guess my brain was somehow well that's a natural thing to do it because we think streaks unlikely we tend to think very small mindedly so if you were just dealing with say four in a row having four heads or four tails in a row is actually in the minority because there's only one way you can get four heads in a row and only one way you can get four tails in a row but we did 20 and so straight after you did four heads or four tails imagine if after that you did head tail head tail head tails there's a whole heap of different permutations you can get which at the beginning have four heads or four tails now that's part of how randomness works is the fact that sure if you're dealing with a small sample you can expect things to be quite mixed up but that's because there's not many permutations head tail head tail tail head tail head is actually you know in the same ballpark number as big streaks but when you're dealing with very long streaks then you could have head head head head head head head and then you've still got all the permutations left to go which is still a huge number it's not just in that position you could get a streak of six heads in a row or six tails in a row at any point and and and if you do that you end up with a whole bunch of permutations which have a streak of six heads or a streak of six tails I've done this with school kids plenty of times and what has happened in my empirical evidence all right is that you do have kids putting in a streak of four heads or four tails but only once they'll only do a maximum once so I actually crunched the numbers and I wanted to work out if how many how many permutations of 20 heads or tails where you had a maximum of three heads or three tails and and just one streak of four it's basically what you've done plus you also allow four heads or four tails now only once that's basically all of this plus one streak of four heads or four tails I really had to crunch the numbers for this I'm telling you so the total number is four hundred and seventy eight thousand five hundred and twenty and that corresponds to forty five point six percent of all permutations it is bigger but it's still less than half so this is kind of the limit of what you're normally mathematically literate person thinks about randomness they actually over over twenty flips of heads or tails will actually cut out half of all permutations as possibilities now half is pretty significant when we're talking about probability right it's like pop you get you get rounded down all the times I do this with a coin I will have a street before exactly exactly more streaks of four in a row and longer streaks it's because I've chosen quite a lengthy number of flips so we've done twenty if it's 16 it doesn't work as much the nature of randomness really presents itself in the length of the number of flips the way we can think about it is how predictable is your sequence really all you need to know is is there a way that you can write your sequence down of heads and tails is there a way to do it that it's simpler and if there is some sort of order or pattern to it then you can actually make it simpler but if your sequence is actually very complicated it's actually harder to write it any smaller than it is and there is a way of categorizing a random sequence by saying the easiest way to write this sequence down is just write it the DNA sequence which makes up me and the fabulous person you are Brady is actually a near random sequence because the DNA sequence contains so much information it actually in a way contains lots of patterns so the way you can think of randomness is it's something that actually has patterns on patterns on patterns on patents and actually fits in very much with the way gamblers think where they say oh I can see a pattern here I can predict what's going to happen next but that's that's just randomness at work randomness shows patterns but there's no logic to it it's not following rules of science or physics or anything it's the way randomness is well know well because you prove to me that that you weren't gonna do it again I think you've started you started big this is basically you did this is like a Hollywood script you start with a big action sequence at the beginning and then it becomes formulaic Thanks well today's episode sponsor brilliant is full of courses and quizzes and puzzles and lots of them touch on the Gamble's fallacy randomness probability in all sorts of ways and this is just a small smattering of the great bounty of stuff you're gonna find on the site which I'm sure you've heard me talk about before it's amazing for mathematics but there's also loads of science and computer science on there as well lots of it's free but you can also get a premium membership that unlocks all sorts of other goodies I know some of you are doing that already but here's something you might not have thought of buying someone else a brilliant membership as 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