This Is NOT 50/50

published on July 9, 2020

Vsauce Kevin here with a homemade deckof 52 mean cards to show you a game thatshould be perfectly fair but actuallyallows you to win most of the time howbecause there's a hidden trick in asimple algorithm that if you know itmakes you the overwhelming favorite even

Though it appears that both players havea perfectly equal 50/50 chance to winwell that's not very fair what is fairwe can consider a coin to be fairbecause it's binary it has just twooutcomes when you flip it

Heads or tails and each of thoseoutcomes are equally probable althoughit could land on its edge in 1993 danielmurray and scott tail posited that anamerican nickel which has a flat smoothoutside ridge could theoretically land

On its edge about one in every 6,000tosses but for the most part since thefirst electrum coins were tossed in thekingdom of Lydia in 7th century BC coinshave been pretty fair as are playingcards like this deck of handcrafted mean

Cards when you pull a card you eitherget a red or a black card crank aartsenit's perfectly binary and there's no wayfor a car to like land on its edge it'seither red or it's black it's 50% like asnap from Thanos given that is it

Possible to crack the theoretical coinflipping code and take advantage of asecret non transitive property withinthis game yes welcome to the humbleNishiyama randomness game but before weget into that look at my shirt I'm

Really excited to announce the launch ofmy very own math designsthis is woven math the launch of my veryown store bridging recreationalmathematics and art this is the pizzatheorem these are concepts that I've

Talked about on vsauce2 like this pizzatheorem or also the Achilles and thetortoise paradox and my goal here is totake cool math concepts actuallyseriously and create soft comfortableshirts that I actually want to wear so

There's a link down below to check themout this is the first drop ever therewill be more to come in the future but Ijust really like the idea of blendingclean sophisticated designs with awesomemath and these shirts just look cool so

That when you wear them people ask whatis that shirt and then you get toexplain awesome math concepts likeAchilles and the tortoise or the pizzatheorem so I think it's great I thinkthat you will to check out the link

Below now let's get back to our gameWalter Penny debuted a simple coinflipping game in the October 1969 issueof the Journal of recreationalmathematics and then Steve humble andYutaka nishiyama made it even simpler by

Using playing cards here's how it worksthe first player chooses a sequence ofthree possible outcomes from our deck ofcards like red black red and then thesecond player chooses their own sequenceof three outcomes like black black red

And then we just flip our red and blackmeme cards and the winner is the onewhose sequence comes up first so in thisexample thanks to guy fieri player onewould have won this game because player1 chose red black red here's a little

Bit of a nitpick because we're notreplacing the red and black cards aftereach draw the probability won't beexactly 50/50 on every draw because eachtime we remove one colored card the oddsof the opposite color coming next is

Slightly higher but it'll never be farfrom perfectly fair and as we play itwill continue to balance out so giventhat each turn of the card has a roughlyequal chance of being red or black andgiven that the likely

Hood of each sequence of three isidentical the probability that bothplayers have an equal chance of winningwith their red and black sequences hasto be 50/50 too right wrong and todemonstrate why I've invited my best

Friend in the whole wide world where areyou best friend Keanu Reeves okay allright you know you're a little tall holdonthat's this okay Keanu will be playerone there are only eight possible

Sequences that Keano can choose red redred red red black red black red redblack black black red red black redblack black black red or black blackblack no matter what Keanu chooses theprobability of that sequence hitting is

Equal to all the other options forplayer one there really is no bad choiceone choice is as good as the next solet's say that Keano chooses black redblack great choice there Keanu now thatI know your sequence I'm going to choose

Black black red okay now we'll just drawsome cards and see which sequenceappears first red Tommy Wiseau some redflexile action so far nobody has anadvantage black where are your fingersOh

Said said Keyon oh you should be sadonce you realize that now there's no waythat I can lose because of having thesetwo black cards in a row even if I pullfive more black cards in a roweventually I will get a red and I will

Win ermahgerdhaha there to his minecraft Steve hassealed the victory for me sorry my mostexcellent dude but I win becauseregardless of what player 1 chooses whatmatters is the sequence that player 2

Picks as player 2 the method here isvery easy I just put the opposite of themiddle color at the front of the lineso when Keano picked BRB I just changedhis middle R to a B and then put back atthe front of my sequence so I just

Dropped the last letter and my sequencebecomes BB arm I'll show you anotherexample if Keanu had shows in red redred then I would have just changed thatmiddle R to a B put that at thebeginning of my sequence drop the last R

And my sequence would be B R are justthat little trick allows me to have anadvantage anywhere from about 2 to 1 upto 75 to 1 which means in the worstpossible scenario for me I win two outof three times and in the best it's

Nearly 8 out of 9 let go right out allthe choice options and their odds asplayer 2 when we apply the algorithmwe're jumping into exactly the rightplace in a cycle of outcomes that player1 doesn't have any control over the best

Player one can do is choose an optionthat's the least bad how is thispossible how can I take something soseemingly fair to both players soobviously 50/50 and turn it so stronglyin my favor the key is in recognizing

That this game is non transitive sothere you go the end wait what istransitive think of it this way you'vegot a B and C a beats b and b beat Ctherefore a beat C because if a beats Band

See then obviously a can beat see thatgame is transitive so if you and yourKeano oops had transitive foodpreferences you'd rather have pizza thantacos and you'd rather have tacos thandog food you'd also rather have pizza

Than dog food simple if you and yourKeanu somehow preferred dog food toPizza then all of a sudden your foodpreferences become non transitive in anon transitive game there is no bestchoice for the first player because

There's no super-powered a insteadthere's a loop of winning choices likerock-paper-scissors in Rock PaperScissors rock which we'll call a losesto paper which we'll call B B is betterthan a but a beats scissors which is C

So a is better than C but B loses to Cso C is better than B and paper B beatsRock a so B is better than a scissors Closes to rock a and beats paper B andwe've got a loop of possible outcomesthat goes on forever with no one choice

Being stronger than the other that's nontransitive since we're in the flowchartmode here's a flow chart thatillustrates the player two winning movesin the humble neshama randomness game soif you follow the arrows you can see

That our BB beats BBB and like BB ourbeats b RB and so forth with the oddsedit you can clearly see how somesequence scenarios go from bad to worsein the humble Nishiyamarandomness variation of pennies game we

Know what sequence of card colors player1 has chosen first so we can jump in themost advantageous part of thenon-transitive loop and make a choicethat gives us a significant advantage byrecognizing that the game is

Non-transitive we can take seeminglyobvious fairness and find a pairtoxic a loophole that nearly guaranteesus success to everyone who doesn'trecognize the entrace civet e it justkind of looks like we're extremely lucky

And why does all of this matter becausebacteria play rock-paper-scissors tomultiply Benjamin Kirk up and MargaretReilly found that bacteria compete withone another in a non-transitive way theyfound that in mice intestines a coli

Bacteria formed a competitive cycle inwhich three strains basically played agame of rock-paper-scissors to surviveand find an equilibrium Penny's game andit's variations illustrate how even ascenario that seems perfectly

Straightforward like unmistakeablysimple should never be taken at facevaluethere's always room to developstrategize and improve our odds if weput in the effort and imagination

Required to understanding the situationand that truly is breathtaking and asalways thanks for watchingyou

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