# Should You Switch? NO! (Here’s why)

published on July 9, 2020

Vsauce Kevin here with two envelopes oneof them contains twice as much money asthe other one and you get to choosewhich one you want to take but beforeyou open your envelope and find outwhether you've won the smaller prize orthe larger prize you have the

Opportunity to switch to the otherenvelope so do you do you switch theanswer is no irrefutable 100 percent noin our scenario it makes absolutely nostatistical difference which envelopeyou choose each one could contain the

Lower amount of money or the higheramount of money with equal probabilityand you have zero information to helpyou choose one over the other rightwrongthe answer is irrefutable 100% yes

Switch the envelopes and there's amathematical explanation for why rightwrong what is happening here let's looklet's let's start at the beginning theorigin of the two envelopes paradox isjust as confusing as the problem itself

In 1953 Belgian mathematician Mauricekrycek posed a scenario in his bookabout recreational mathematics in whichtwo men compared the values of necktiestheir wives bought them with neither manknowing the prices also in 1953 a math

Book credits a similar puzzle usingplaying cards to physicist ErwinSchrodinger ever hear of Schrodinger'scat where a cat in a box is both aliveand dead until you open the box and findout yeah same God in an hour problem the

Forces at play are the same whetherwe're using playing cards neckties orenvelopes oh also I'm wearing mybrand-new birthday paradox shirt I hadthe floating space baby bracketed by thechart

Detailing the probability of at leasttwo people sharing a birthday versus thenumber of people it's in my new vsauce2store link below where it for all thespace children so look the logic for notswitching envelopes is pretty

Straightforward given that you don'tknow anything about either one or evenwhat the higher and lower values areinside them one is just as good as theother but the math in favor of switchingis kinda compelling let's say that the

Envelope you choose contains X dollarswe don't know how much X is just thatthere's something in the envelope we cancall X the probability of that envelopecontaining the smaller value is 1/2 andthe probability of it containing the

Larger amount is as 1/2 it's 50% theother envelope either contains twicethat amount 2x or half that amount 1/2 Xif X turns out to be the higher valuethan the other envelope contains 1/2 Xif your envelope X is the lower value

Then the other envelope contains 2x makesenseI've talked about expected value in afew other vsauce2 videos it's theaverage of the possible outcomes of aseries of probabilistic events which we

Can use to identify the mostadvantageous course of action so to findthe expected value of the envelope wedidn't choose we add the value of theonly two possibilities weighted by theirprobability like this 1 over 2 times 2x

Plus 1 over 2 times 1/2 x equals 5 over4 x half of the time the other envelopewill contain 2x and half of the time theother endwill contain 1/2 X which equals anexpected value of 5/4 X 5/4 X is 25%

More than just X so our best possiblechoice is to switch to the otherenvelope cool we did it but that makesno sense you have a simple clean 50-50chance of choosing the envelope withmore money inside and switching can't

Change that but the math just told usthat it can don't worry though it getsweirder what if I said you could switchenvelopes again if you wanted to wellthen you just go through the sameprocess over and over again always

Switching never opening forever and everand ever and ever and nerver and blurver those aren't words but you get thepoint because switching makes the mostmathematical sense with switchinggaining you a 25% advantage and I just

Proved itlet's come back to math later it's timefor logic logician Raymond Smolyan posedtwo scenarios for the paradox first therisk of gaining twice the amount in yourenvelope is worthwhile when the

Alternative is only losing half of ityou're gaining X or you're only losinghalf of X the potential payoff is twiceas high as your risk of loss makesperfect sense but one envelope containsX and the other contains 2x so you're

Either gaining X or losing X also makesense but both can't be true so we'regonna do this problem again usingeconomics professor Barry nail buffsvariation let's say that we give anenvelope to player 1 and then flip a

Coin if it's heads we put double thatamount in the second envelope and giveit to player 2 if its tail's we put halfthat amount in the envelope and give itto player 2 neither player knows howmuch is in their envelope or the outcome

Of the coin toss once they open theirenvelopes in secret they can switch witheach other if they both agree to switchit player 1 opens their envelope andfinds ten bucks with equal probabilitythey think that switching to player

Two's envelope will get them 20 bucksfor a \$10 gain or take them down to 5bucks for a 5 dollar lossplayer one decides to switch becauseswitching gives them a greater possiblereward relative to what they might lose

Player twos in another room opens theirenvelope and finds the five bucks theythink there's a 50% chance thatswitching to player ones envelope willget them 10 bucks for a five-dollar gameor get them only two dollars and fifty

Cents for a two dollars and fifty centloss so player 2 also thinks that whatthey stand to gain is greater than whatthey might lose they are absolutely surethat their upside is better than theirdownside but it's obvious that both

Players can't have an advantage in thisgamethat's impossible if you're confused atthis pointgood Martin Gardner one of history'sgreatest puzzle solvers and paradox

Explainers knew the answer but admittedthat there just wasn't a simple way tocommunicate the flaw in mathematicalreasoning and sadly that's where ourjourney ends no it's not because we canfigure this out think of it this way the

Total sum of money in the envelopes isthree units a small unit and then alarger value that's made up of twosmalls that combine to form a total of Xif you've got these smaller envelope itsvalue is x over 3 and if you have the

Larger envelope its value is 2x over 3when you switch from small to largeyou've gained X over 3 and when youswitch from large to small you lose Xover 3 so the expected value ofswitching is 1/2 times 2x minus x over 3

Plus 1/2 times X minus 2xover three which equals zerothere's just no value in switching itdoesn't help us and it doesn't hurt uswhich is probably what you knew until wegot math involved the two envelopes

Problem is a lot more than just X and 2xwhich is what makes it a false iticalparadox because there appears to be again from switching when we fail toconsider that the states of the twoenvelopes don't actually reflect simple

Values of x + 2 X a to X envelope isonly 2 X when it's the larger amount andthe 1/2 X envelope is only 1/2 X whenit's the smaller amount withoutrecognizing that difference basicalgebra playing off our common sense

Fools us into thinking that each switchgives you a gain of 5 over 4 or 25% it'sreally a matter of perspective dependingon which envelope you're holding whichone you think you have which one youthink you don't have and which one you

Want to wind up with your course ofaction might be incredibly clear or notand then clear again and then not againuntil you realize that no matter if youdiagnose it mathematically no matter ifyou deconstruct it logically no matter

What you do when it comes to the twoenvelope paradox you were right allalongand as always thanks for watchinghey I actually started developing thisvideo and began shooting it when

Brilliant org came to me aboutsponsoring it I really like working withbrilliant because it's an entireplatform based on learning math andscience by playing with them somethingyou may have noticed I'm kind of a fan

Of and surprise surprise they have areally good explanation of theseperplexing probability scenarios they'reinteractive puzzles help you exposemisconceptions and you are learning byplaying this is just one of 60 plus

Courses on brilliant that teach you bywalking you through puzzles and guidingyou in figuring out the solutionsbasically teaching you how to thinkwhich I would highly recommend so headover to brilliant org slash vsauce2 and

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