Vsauce!
Kevin here, with 64 kittens, 2 wallets, 1 coin -- and some math that lies to you.
Some of you are gonna be okay with that, because you’ll already have found cognitive shelter
in the unknown.
Others won’t even see it.
I’ve made a lot of videos about different probability problems and paradoxes -- Russian
roulette, Monty Hall, the Birthday Paradox -- and… we’ve solved them all.
But what happens when the math is right and there’s still a paradox?
What happens when you KNOW a right answer is actually wrong?
Wallets.
Imagine I choose two people randomly as they walk on the streets of Manhattan.
I give them the option to play a game.
They compare the amount of money in their wallets, and the person with the LOWER amount
wins all the money in the other wallet.
Does either person have an advantage?
The math says NO.
YOU say YES.
But wallets can wait.
Let’s start with coin flips, the probability of which is perfectly quantifiable.
I’ve got a coin with YES on one side and NO on the other.
The odds of getting a YES or NO are 50/50… other than the 1 in 5,000 chance that it lands
on its edge.
So 50/50.
The odds of getting YES / NO / YES / NO / YES are .5 raised to the power of 5, or .5 x .5
x .5 x .5 x .5 = .03125.
Cool.
The odds of getting 5 YES’s or NO’s in a row?
WELL!
Now.
That’s… also .5 raised to the power of 5.
Every combination of YES and NO has exactly the same chance of occurring.
But you KNOW deep down that getting 5 in a row doesn’t happen nearly as often as a
mix of the two.
How can two things have the exact same probability of occurring, yet occur less often?
I can explain….
WITH KITTENS.
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You have a litter of kittens (or your cat does) -- let’s say the cat has 4 kittens,
and there’s a 50/50 chance of each kitten being a boy or a girl.
Every combination of kittens is .5 raised to the power of 4, or .0625 -- but the most
common combination isn’t the equal split of 2 boys / 2 girls you’d expect.
It’s actually 3 of one and 1 of the other.
When you map out all the possible combinations of kittens, there are 16 combinations that
all have the same exact probability of happening -- but getting a perfectly even 2 of each
happens in just 6 of the 16 combos, or 37.5%.
A 3-1 split happens in 8 of the 16 combinations.
50% of the time you’ll get 3 and 1, despite the 50/50 math telling you everything’s
got the same chance.
Simple math tells you one thing, but combinatorics using that math tells you another.
Okay.
Let’s go back to wallets.
Do you have an advantage in this game?
If your wallet has less money in it, you’ll always win MORE than that amount by taking
the other wallet that has more money in it.
If YOUR wallet has more money, you’ll only lose what you’ve got.
Since the game is totally random and neither player knows anything about the other, the
probability of each player winning and losing is the same -- but when you win, you’ll
get more money than you LOSE when you DON’T win.
So you’ve got a major advantage in this game.
But when the other player loses, they’re only losing what’s in THEIR wallet.
And when they win, they’re winning MORE than what’s in their wallet.
THEY’VE got a major advantage in this game.
The question is: How can two players competing against each other BOTH have an advantage?
Here’s how…
THEY CAN’T!
If there are two outcomes -- X and Y -- we can assign probabilities to each, and adding
those probabilities together HAS TO add up to 1.
Having an advantage would mean a probability greater than .5… if X and Y are both greater
than .5, you’re gonna get more than 1 when you add them -- and that’s probabilistically
impossible.
It would be like betting on a sports game where Vegas says BOTH teams are favored to
win.
It makes no sense.
And look.
Maurice Kraitchik’s payoff matrix for the wallet game shows that the two players have
exactly the same probabilities of winning.
It really is 50/50, and the math proves it.
But the logic of each player is… flawless.
It’s unassailable.
Neither player CAN have an advantage, but both players have perfectly reasonable logic
suggesting each one DOES have an advantage.
The problem here is that you NEED to know, and discarding information makes it easier.
I’ll talk more about this in a future video -- but think how much easier this problem
gets when you just trust the math, or you go with logic and your gut.
Reducing a complex scenario between math and reality lets you eliminate the paradox and
proceed comfortably.
But when faced with explaining the faulty logic of these two wallets, some of the greatest
math minds could only proceed comfortably knowing that they... couldn’t really tell
ya.
Kraitchik first proposed the problem in 1944 -- and The Two Envelopes Paradox was conceived
as a derivative of it in 1953.
Nearly 80 years later, we don’t have a clear answer to why the math works and the reality
doesn’t.
You can factor in discrete distributions, random variables, or continuous density functions
to get deep into the math -- but the TRUTH is that probability tells you one thing, and
logic tells you another.
The moment we know something about the players in the wallet game, we enter the territory
of the subjectivistic interpretation of probability -- the Bayesian notion of utilizing information
about a problem to assign possible values to different frequencies of outcome.
So like, if one player is 18 years old, you might think they’re less likely to have
a lot of money in their wallet because younger people are more likely to earn less, carry
cards or pay with their phones.
If a player is older and on vacation in Manhattan, their wallet might be more likely to be LOADED
with cash.
But in our wallet game, we don’t know ANYTHING.
We have math that proves it’s equal and logic that proves it isn’t.
We have such a strong tendency toward equiprobability that when logic tells us we’ve got an exception
to a normal 50/50 rule, it shocks us into taking it too seriously.
It messes with our uniformity belief enough to make us trust it.
Martin Gardner wrote that Kraitchik was “no help” about logic in the wallet paradox.
Gardner couldn’t explain it, either.
All these papers by very smart mathematicians get into the math but just discard the very
real logical objection.
Often by concluding with something along the lines of, "uhhh... don't play this game!"
No one can explain why without making a few assumptions.
Maybe the wallet game is just an antinomical paradox that we have to live with.
But the coin flips and kittens?
That’s the type of math that can mislead you.
Probability just isn’t frequency -- and frequency is reality.
So life may be a series of 50/50’s, wins and losses, yeses and nos -- that lead us
all to wildly different outcomes.
Like sitting at a whiteboard table with 64 kittens and a pink leopard print wallet.
And as always -- thanks for watching.