Vsauce! Kevin here, with a homemade deck of 52 meme

cards to show you a game that should be perfectly fair… but actually allows you to win most

of the time. How? BECAUSE. There’s a hidden trick in a simple algorithm

that if you know it, makes you the overwhelming favorite even though it appears that both

players have a perfectly equal 50/50 chance to win. Well, that’s not very fair. What is fair? We can consider a coin to be “fair” because

it’s binary: it has just two outcomes when you flip it, heads or tails, and each of those

outcomes are equally probable. Although… it could land on its edge… in

1993, Daniel Murray and Scott Teale posited that an American nickel, which has a flat,

smooth outside ridge, could theoretically land on its edge about 1 in every 6,000 tosses. But for the most part, since the first electrum

coins were tossed in the Kingdom of Lydia in 7th century BC, they’ve been pretty fair. As are playing cards, like this deck of hand-crafted

meme legends. When you pull a card, you get a red or a black

card. Crying Carson. It’s perfectly binary, and there’s no

way for a card to like, land on its edge. It’s either red or it’s black. It’s 50% like a snap from Thanos. Given that, is it possible to crack the theoretical

coin-flipping code and take advantage of a secret non-transitive property within this

game? Yes. Welcome to the Humble-Nishiyama Randomness

Game. But before we get into that. Look at my shirt! I’m really excited to announce the launch

of my very own math designs. This is Woven Math. The launch of my very own store bridging recreational

mathematics and art. This is the Pizza Theorem. These are concepts that I’ve talked about

on Vsauce2 like this Pizza Theorem or also the Achilles and the Tortoise paradox. And my goal here is to take cool math concepts

actually seriously and create soft, comfortable shirts I actually want to wear. So there’s a link below to check them — this

is the first drop ever there will be more to come in the future — but I just really

like the idea of blending clean sophisticated designs with awesome math. And these shirts just look cool so that when

you wear them people ask, “What is that shirt?” and then you get to explain awesome math concepts

like Achilles and the Tortoise or The Pizza Theorem. So I think it’s great, I think that you

will too, check out the link below now let’s get back to our game. Walter Penney debuted a simple coin-flipping

game in the October 1969 issue of The Journal of Recreational Mathematics, and then Steve

Humble and Yutaka Nishiyama made it even simpler by using playing cards. The first player chooses a sequence of three

possible outcomes from our deck of cards, like red, black, red. And then the second player chooses their own

sequence of three outcomes. Like black, black, red. And then we just flip our red and black meme

cards and the winner is the one whose sequence comes up first. So in this example, thanks to Guy Fieri, player

one would’ve won this game because player one chose red, black, red. Here’s a little bit of a nitpick. Because we’re not replacing the red and

black cards after each draw, the probability won’t be exactly 50/50 on every draw — because

each time we remove one colored card, the odds of the opposite color coming next is

slightly higher — but it’ll never be far from perfectly fair, and as we play it will

continue to balance out. So given that each turn of the card has a

roughly equal chance of being red or black, and given that the likelihood of each sequence

of three is identical, the probability that both players have an equal chance of winning

with their red-and-black sequences has to be 50/50, too, right? Wrong — and to demonstrate why I’ve invited

my best friend in the whole wide world. Where are you best friend? Keanu Reeves. Ah, alright. Keanu, you’re a little tall. Hold on. How’s this? Okay, Keanu will be player 1. There are only 8 possible sequences that Keanu

can choose: RRR, RRB, RBR, RBB, BRR, BRB, BBR, and BBB. No matter what Keanu chooses, the probability

of that sequence hitting is equal to all the other options. For player 1, there really is no bad choice,

one choice is as good as the next. So let’s say Keanu chooses BRB. Great choice there, Keanu. Now that I know your sequence, I’m going

to choose BBR. Okay, now we’ll just draw some cards and see

which sequence appears first. Red, Tommy Wiseau. Some red Flex Seal action. So far nobody has an advantage. Black, where are you fingers? Uh oh. Sad, sad Keanu. You should be sad once you realize that now

there’s no way that I can lose. Because of having these two black cards in

a row, even if I pull five more black cards in a row, eventually I will get a red and

I will win. Ermergerd. Ah hah. There it is. Minecraft Steve had sealed the victory for

me. Sorry, my most excellent dude. But I win. Because regardless of what player one chooses,

what matters is the sequence that player 2 picks. As player 2, the method here is very easy

— I just put the opposite of the middle color at the front of the line, so when Keanu picked

BRB, I changed his middle R to a B, and then put that in the front of my sequence. So I just dropped the last letter and my sequence

becomes BBR. I’ll show you another example. If Keanu had chosen red, red, red. Then I would’ve just changed that middle R

to a B, put that at the beginning of my sequence, drop the last R, and my sequence would be

BRR. Just that little trick allows me to have an

advantage anywhere from about 2 to 1 up to 7.5 to 1. Which means in the worst possible scenario

for me, I win 2 out of 3 times. And in the best, it’s nearly 8 out of 9. Look, I’ll write out all the choice options

and their odds. As Player 2, when we apply the algorithm we’re

jumping into exactly the right place in a cycle of outcomes that Player 1 doesn’t

have any control over. The best Player 1 can do is choose an option

that’s the least bad. How is this possible? How can I take something so seemingly fair

to both players, so obviously 50/50, and turn it so strongly in my favor? The key is in recognizing that this game is

non-transitive. So there ya go.The end. Wait… What is transitive? Think of it this way: you’ve got A, B, and

C. A beats B, and B beats C. Therefore, A beats C. Because if A beats B and B beats

C then obviously A can beat C. That game sequence is transitive. So like if you and your Keanu had transitive

food preferences, you’d rather have Pizza than Tacos, and you’d rather have Tacos

than Dog Food. You’d also rather have Pizza than Dog Food. Simple. If you and Keanu somehow preferred Dog Food

to Pizza, then all of a sudden your food preferences become non-transitive. In a non-transitive game, there is no best

choice for the first player because there’s no super-powered A. Instead, there’s a loop

of winning choices… like rock, paper, scissors. In rock paper scissors, rock — which we’ll

call A — loses to paper, which we’ll call B. B is better than A. But A beats scissors,

which is C. So A is better than C. But B loses to C, so C is better than B, and

paper B beats rock A, so B is better than A. Scissors C loses to rock A and beats paper

B — and we’ve got a loop of possible outcomes that goes on forever, with no one choice being

stronger than the other. That’s non-transitive. Since we’re in the flow chart mood here’s

a flow chart that illustrates the player 2 winning moves in the Humble-Nishiyama Randomness

game. So if you follow the arrows you can see that

like RBB beats BBB and like BBR beats BRB. And so forth. With the odds added, you can clearly see how

some sequence scenarios go from bad to worse. In the Humble-Nishiyama Randomness variation

of Penney’s Game, we know what sequence of card colors player one has chosen first,

so we can jump in the most advantageous part of the non-transitive loop and make a choice

that gives us a significant advantage. By recognizing that the game is non-transitive,

we take seemingly-obvious fairness and find a paradoxical loophole that nearly guarantees

us success. To everyone who doesn’t recognize the intransitivity,

it just kinda looks like we’re extremely lucky. And why does all of this matter? Because bacteria play rock paper scissors

to multiply. Benjamin Kirkup and Margaret Riley found that

bacteria compete with one another in a non-transitive way. They found that in mice intestines, E. coli

bacteria formed a competitive cycle in which three strains basically played a game of rock

paper scissors to survive and find an equilibrium. Penney’s Game and its variations illustrate

how even a scenario that seems perfectly straightforward, like unmistakably simple, should never be

taken at face value. There’s always room to develop, strategize,

and improve our odds if we put in the effort and imagination required to understanding

the situation. And that truly is…breathtaking. And as always — thanks for watching.

I can't sell the meme cards but I hope you enjoy Woven Math! Wrap your body in sweet, sweet knowledge. https://represent.com/store/vsauce2

Real question is how to win if you have to choose the sequence first

@ Vsauce2What if you WANT to loose? (for whatever reason…?)

Do I take the middle, flipp it & put it at the back instead (of the front)? (I think: No. It's probably some other combination? 7:33 try to

consistentlytransform from middle- to left- -column )10:42 intransitivity not intransivity

VSauce2 next upload: A Poker Game You Can Always Win

Next up: Grime dice!

You can beat most people at Rock Paper Scissors like this:

Tell them that you want to do a one round Rock Paper Scissors.

Tell them you are going to win. That you know that you will win.

If they’re confused they should throw scissors. You just throw rock to counter scissors.

It's like when you pick your starter in Pokémon, and then your rival picks the starter strong against yours. (although nowadays, Pokémon rivals pick the starter weak to yours)

Although you can catch other Pokémon that are strong against your rival's starter, and other Pokémon that are strong against your rival's team.

what if ur friend gets suspicious and let's u pick first

Loving that Dwight Schrute coin

Thanos card

Thanos card

So I gotta flip a coin 6000 to land it on its edge

There is a 50 percent chance to get it on the first 3000 tries

I have two hands

If I flip the coin with both hands, there is a 50 percent chance to get it in the first 1500 tries

If I take a fistful of 30 coins in both hands, totaling as 60 coins, there is a 50 percent chance to get it within the first 25 tosses

If I get 50 people doing this exact thing with me, there will be a coin on its side.

Please bring back mind blow I miss gaining knowledge from the world of science and it’s kinda nostalgic😌

Is there any possible way to win such a deck xdd

Tacos are better than pizza.

9:56 is like https://youtu.be/HuB_Pm4Xxbs?t=7

A fair game that is unfair

Booo at 11:42 I thought you were gonna say "and that truly is… excellent!"

I am in awe of the way in which you managed to fool everyone that a game in which the second player gets to decide after he knows what the first player chose is fair. It's not fair if one player chooses first every time, even if you didn't know the probabilities and stuff.

But hey, this is a mind excercise, not a tutorial of a competitive game.

Great video, cheers!

ooooo, you’re going to get into long term meta stability in non-transitive games, aren’t you? 😘

Give me that deck

why did you change the title

Hey Vsauce2 can you make a video on Baccarat the game? I think it'll be sick if you figure out a way to beat it

A bit dull

SANS

no you're breathtaking

If you lose you get to go first the next round

For the longest time, I called it "Paper, Rock, Scissors"… I guess I was wrong.

Keanu was too tall, so Kevin transformed him into a Youtuber

what is stopping player 1 from choosing brb making player 2 pick bbb? wouldn't that make player 1 win more being that player 2 has to get three in a row?

Sans is the joker

Eheheheheheheh

Who got triggered that he didn’t say “Perfectly balanced, as all thing should be.” When he showed Thanos.

Mans just said that his best friend was KeanO Reeves. How can you not know his name by now?

Plot twist

Your friends says that he wants to go second or else he's not playing.

10:02 Brb! 😀

"Heads"

Heads"Or tails"

HeadsAll vsause two videos are merging into one

Nawww change of title

So this game is essentially like playing Rock Paper Scissors but you get to see what they pick first and then you pick

How do the odds change if you use red/diamond, red/heart, black/spade & black/club?

my best friend — said Vsauce

later smacks him off the table

Jake: "We don't sell these cards."

Vsause member's hearts: 9 of spades

Us: How we wish we could summon the power of Jack of spades

I've only just found your channel, I watched the sprouts one before this one. I remember that from way back when but it never really interested me until I watched your vid. And being a cardist, naturally I was eager to see this one. And I've been sat playing with myself(excuse the pun) chuckling strangely watching the outcomes. Time to win some beer… Lol. Loving the channel, +1 subscriber

5:25

keanu reeves: be right back

But if BBR beats BRB 2:1, then RRB should beat RBR 2:1 too, not 3:1 as you said/drew/wrote , no?

Woah

Pokemon starter types are non-transitive

Are those Whang merch playing cards?

ke ya nu it's keanU not keanO

6:26 YOU HAVE COMMITED A MORTAL SIN.

Great video as usual, the last probability written, at 10:07, is wrong I think though, it should be 2:1, shouldn"t it?

I'm a bit late, but I turned it to a program. for each of the 8 possibilities it plays 1,000,000 games, here is how the results turned out:

PS. You will be playing the second (right color), wins are counted for that one.

[RRR] –> [BRR]

WIN: 875,346

LOSE: 124,654

WIN RATIO: 87.53%

———-

[BBB] –> [RBB]

WIN: 875,328

LOSE: 124,672

WIN RATIO: 87.53%

———-

[RRB] –> [BRR]

WIN: 749,831

LOSE: 250,169

WIN RATIO: 74.98%

———-

[BBR] –> [RBB]

WIN: 750,706

LOSE: 249,294

WIN RATIO: 75.07%

———-

[RBB] –> [RRB]

WIN: 667,323

LOSE: 332,677

WIN RATIO: 66.73%

———-

[BRR] –> [BBR]

WIN: 666,546

LOSE: 333,454

WIN RATIO: 66.65%

———-

[BRB] –> [BBR]

WIN: 666,961

LOSE: 333,039

WIN RATIO: 66.70%

———-

[RBR] –> [RRB]

WIN: 666,937

LOSE: 333,063

WIN RATIO: 66.69%

why did you changed the title?

So you’re just trying to beat the opponent to the possibility of their series of occurring.

You… look like…. Keanu Reeves…

If anyone gets the reference then they are my best friend.

Hey, this isn't Scam School!

bro, i love this guy. he sounds like Hickup from How to Train Your Dragon (no offence btw)

Why not just write it both at the same time so it is fair?

When I don't understand ☹️

great looking merch! might get myself one!

I have a QUESTION, if nothing is faster than light how did the dark get there first? Random thought.

Cry me carson

I've seen a coinflip that landed on the edge, we were all shocked

Stop de video in 0:05

Do you know da we :V XD

What the hell mens I need those cartds jajajaja XD

… I'm disappointed the Knuckles isn't the Q.

I’m still confused

Nothing beats rock moron

What happens if you extend the game to four choices?

You should make a video called “Or is it?”.

I was gonna get my bro some meme cards for Christmas 😭

I have tosed a 1swedish kr and I have done it with a clear red dise with sharp corner and it stood fully still but my French tired it 😭

In Sweden it is rock bag scissor because a bag can have a stone in it but a A4 papper sheet can't

The title should have been called “The Card Game You Can (almost) Always Win, ft. Keanu Reeves.”

Qurikology2 confirmed

i still don't understand "why this works though. you made the joke "it's non-transitive, the end," but it still feels like you didn't actually explain it. you still just kind of said, "it works because it does."

for example, why are the odds the way they are?

Thanos is Queen10:05 Suspiciously Swastik-ish

Didn't mention how there are advantages in rock paper scissors due to frame data and other stuff. Rock is the fastest and generally will win 80% of first strikes. Paper beats the all powerful rock and is a great counter pick a lot of the time. Scissors is weak as can be, low frame data, and should only be used as a mixup. I managed to beat a group of ten people in janken twice in a row utilizing this, not even rock paper scissors is entirely fair.

Love it. But the flow chart @10:05 is incorrect. RRB beating RBR should be equal to BBR beating BRB.

Meme cards it's just a deck of pics of vsause

When he said my best friend I thought he would pull out the Yoda that has a heart attack

0:30 insert “ and how much does it weigh” here.

Turn your speakers up and listen for the mysterious click at 10:01

A L M O S TMe skipping ahead:

Video: to Thanos

Me: come again?!

Be right back

Back

What was the point of the coin landing on its edge tangent

I missed the whole video because I was distracted by those amazing card omg I neeeed them

vsauce – makes the memes

vsauce2 – uses some memes

vsaucce3 – a lil bit of both

U MISSED AN OPPORTUNITY

u could have explained the 8 possible sequences and put the bois to be an example for B B B

and say "oh look its me and the b's"

I was hoping for a bit more depth with the mathematical reasoning! But good video regardless.

I see sans on a black card!

Me having a conversation with Kevin

Vs2: you like pizza, right?

Me: yeah I love pizz…

Vs2: WRONG!

Me: err… ok then.

Vs2: left isn’t right left is left right

Me: wha…

Vs2: Wron…

Me: RIGHT!

Vs2: I need to call Michal.

This is great to know. A week later my gf hates me now.

smh no p3pe

Ah Penney's game (who is from Numberphile?)

Shouldn't it be 7-to-1 for BBB? If we get a single red we lose. So 1/2 * 1/2 * 1/2 = 1/8 vs 7/8 so a 7-to-1 chance

0:57 isnt it a. Five baht coin form thailand

Pokémon is non-transitive, fire beats grass, grass beats water, but water beats fire… MY CHILDHOOD KEVIN, MY CHILDHOOD!

This is just a reminder that memes are hieroglyphic.