# The Easiest Problem Everyone Gets Wrong

vsauce kevin here and i am about to stretch my winky until it snaps actually i've got three of these anti-stress emoji people called winkies only one will survive and you will witness an incredibly simple

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that almost## everyone

### gets

#### wrong

and at the end get to the bottom of one of the most famous probability#### problem

s ever it's time to meet the winkies orange purple and blue they've been sentenced to death but just one will be pardoned the other two are walking the green milebut they don't know which two it will be all they know is that i warden kevin get to decide right now each winky's chance of survival is one out of three so one over three plus one over three plus one over three equals one with every winky having an equal shot at living there's no paradox here yet the fairest least biased way to choose which winky lives is to put three slips of paper into a hat and randomly draw a color and mix around your slips and here we go but what if they know

that i drew a color yet i don't tell them which one it is these winkies would be left to wonder who will live who will die and what their own personal odds of survival really are kevin kevin kevin you can tell me i can keep a secret tell me who will live my winky is talking to me and i think i recognize that voice a keeper of internet history how'd you end up on death row anyway i can't tell you who survives it's it's just not fair and i definitely can't tell you whether

you live or die i understand you bear a heavy burden just tell me one of the winkies that will be executed if purple lives tell me blue if blue lives tell me purple and if i'm going to live just choose which of the other two you'd like to name do this for me kevin please i don't i don't know orange winky it really doesn't seem fair please given pretty please you are so handsome and those glasses look so good on you okay what's the harm i can't possibly give orange

winky any useful information here okay blue will be executed right now i didn't mean to put his body back that way upside down but you know what warden kevin is uh is a cruel warden kevin you fool now i know my odds are 50 50. i will live or purple will live there is no alternative you stupid stupid man oh he's so dumb and also fat oh no he's right i got tricked by a winky now there are only two choices one will live and one will not so his odds improved from one out of three to

fifty fifty right

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orange winky you still only have a one out of three chance to live and purple winky is actually twice as likely to win the pardon this may remind you of the monty hall#### problem

and we'll get to why that's important later but for now let's break down the only possible scenarios that orange winky faced remember that i can't tell orange winky whether he lives or dies when i named blue we can compare the two possible outcomes after naming blue scenario onewhere i name blue and purple lives has a one out of three probability of occurring scenario three where i name blue and orange lives has a one out of six chance of happening since one over three is twice as large as one over six and since they're the only two possible outcomes after naming blue meaning their probabilities have to add up to one purple actually has a two out of three chance of survival and orange only has a one out of three chance of avoiding execution the probabilities of two

events that appear to be equally likely just aren't and we can't handle that all right it's time to reveal which wiki will live and which winky will die this slip of paper determines our survivor and purple lives sorry orange kevin time for you to no we have a message become history my body is stretching do not scratch me kevin no was it because i called you fat i am dead welcome to the three prisoners

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which is actually the bertrand's box paradox and kind of the montyhall

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but mostly the bertrand's box paradox which came out the same year the eiffel tower opened van gogh painted the story night and nintendo kopai first started producing playing cards they eventually made video games in 1889 70 years before recreational math columnist martin gardner started playing mind games with prisoners on their execution date french mathematician joseph bertrand published calculate probability in which he posed a paradox that became the basis the genesis formonty hall and will require me to stick my winkies in pudding let me explain after making me a plate of tendees in my favorite dipping sauce my mom secretly hid winkies in these three large candy dishes full of pudding one of them contains two winkies of the same color another one contains two winkies of a different color and then one contains one winky each of those two colors i'm pretty sure that mom said one winky color is yellow but i'm not sure what the other one is this is so much

grosser than i thought it was going to be let's reveal this winky's color oh thank you for licking me clean kevin your tongue is so smooth how are you alive you can't kill me kevin we're friends now okay well clearly we have an orange winky somehow back from the dead and we know that the second winky color is yellow but what are the chances that the other winky in this pudding is also orange we know that the three dishes are orange orange orange yellow and yellow yellow each

combination is equally likely to be chosen at the start we also know that this can't be the yellow yellow dish because i just pulled out the creepiest orange winky on the planet that means it's coming from orange orange or from orange and yellow and both had the same chance of being selected once we pull the orange winky we know that the odds the other winky in this dish is orange is 50 50 because i've clearly chosen either the orange orange pudding or the orange yellow pudding right

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again for the same mathematical reason as our prisoner#### problem

this is the mathematical foundation of all these scenarios our inability to recognize that what appears to be 50 50 really isn't hold on before i explain further let's just find out real quick where the winkies are because i actually don't know well this was orange orange ah so it was orange orange jar number two i can't even tell yeah okay that's yellow now is this one orange or is this the yellow yellowjar time to find out looks orange looks orange let's see yep there's the other orange so that was our yellow orange which means that this has to be the yellow yellow yellow yellow i already have to clean up this pudding or else we can't continue the video so i'll be right back all right now that the vsauce mukbang portion of our video is complete let's get to the root of this

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bertrand conceived his original paradox with gold and silver coins in boxes he had the samebasic setup we used gold gold gold silver and silver silver with the player reaching into a random box drawing a gold coin and wondering about the odds of that box containing another gold coin once we pull a gold coin or an orange winky or one of the prisoners to be executed it feels like our odds of whatever comes next are an equal 50 50. we're just choosing between two leftover options that appear to be equally likely we know the other coin is either gold or silver but look if you pull a

gold coin you either pull gold coin one gold coin two or three and at this point you conceptually know that you're not dealing with the box with the two silver coins you don't know exactly what it is but you do know what it's not if you choose gold one the other coin is gold two if you choose gold two the other coin is gold 1. if you choose gold 3 you'll get silver one so 2 out of 3 times you draw gold it'll be paired with another gold and only one time out of 3 will you get

stuck with silver but it's almost impossible to shake the notion of two equivalent options even when the math shows us that it's

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it's the same reason why the monty hall#### problem

is a#### problem

the prisoner#### problem

and monty hall add a story to the math and an additional third party it seems like that would make it more complex but it actually helps you understand the probability more easily adding a second player's perspective lets us see that the warden will never reveal wholives and monty hall will never reveal the door with the money despite that additional information we still get it

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why is it so hard when the solution is right in front of our face probably because it is right in front of our face and it's why very very smart people often get it more#### wrong

than right i mentioned this in my what is a paradox video but when marilyn vos savant wrote in parade magazine that switching doors in the monty hall#### problem

resulted in a two over three chance ofsuccess instead of being an obvious 50 50 trade parade received over 10 000 letters telling her she was

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about 1 000 of them were signed by phds and she claims many were sent on official letterheads from college math and science departments and i still get comments on my video from people all over the world telling me that i'm#### wrong

massimo piatelli paul marini author of inevitable illusions said that no other statistical puzzle comes close to fooling all the people all the time andresearch in 1995 by granzberg and brown found that only 13 of people correctly switched monty hall's doors in a 2003 paper in the journal of experimental psychology stephan krauss and xt wang described these particular

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s from bertrand's box to monty hall as cognitive illusion stating that they demonstrate people's resistant deficiency in dealing with uncertainty they found that only three percent in the control group solved the#### problem

with mathematically correct reasoning andthat a significant

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was perspective when participants were able to think of the#### problem

from the perspective of monty hall instead of the game show contestant or thinking as me warden kevin instead of as the prisoner orange winky their chances of choosing the right pathway improved vaulting them from simple non-bayesian thinking into more complex bayesian analysis in the 1990s it appeared that there was no way to break the mathematical illusion of bertrand's box the three prisoners orthe monty hall

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explaining the math just didn't seem to work it turns out the math isn't the#### problem

we are uncertainty is hard for us changing perspectives is harder you're watching this video because you're a smart curious person and by altering your perspective you don't just dismiss a mathematical illusion that has perplexed us for over a century because you've identified its flaw you don't just solve it you embrace it you absorb it you understand it andyou ultimately use it to your advantage you know it and i know that i have so much pudding to eat and as always no thanks for watching pudding i mean surprise you