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Vsauce! Kevin here, with two envelopes. One
of them contains twice as much money as the
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other one, and you get to choose which one
you want to take.
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But before you open your envelope and find
out whether you’ve won the smaller prize
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or the larger prize, you have the opportunity
to switch to the other envelope... so, do
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you? Do you switch?
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The answer is… No. Irrefutably, 100% no.
In our scenario, it makes absolutely no statistical
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difference which envelope you choose. Each
one could contain the lower amount of money
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or the higher amount of money with equal probability,
and you have zero information to help you
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choose one over the other. Right?
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Wrong. The answer is irrefutably, 100% yes
-- switch the envelopes. And there’s a mathematical
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explanation for why. Right?
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Wrong.
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WHAT IS HAPPENING HERE!?
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Let’s...let’s... let's start at the beginning.
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The origin of the Two Envelopes Paradox is
just as confusing as the problem itself. In
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1953, Belgian mathematician Maurice Kraitchik
posed a scenario in his book about recreational
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mathematics in which two men compared the
values of neckties their wives bought them,
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with neither man knowing the prices. Also
in 1953, a math book credits a similar puzzle
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using playing cards to physicist Erwin Schrodinger
-- ever hear of Schrodinger’s cat? Where
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a cat in a box is both alive and dead until
you open the box and find out? Yeah, same
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guy -- and in our problem, the same forces
at play are the same whether we're using playing
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cards, neckties, or envelopes.
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Oh also, I’m wearing my brand new Birthday
Paradox shirt. I have a floating space baby
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bracketed by the chart detailing the probability
of at least two people sharing a birthday
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vs. the number of people. It’s in my new
Vsauce2 store -- link below. Wear it, for
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all the space children.
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So look. The logic for not switching envelopes
is pretty straightforward. Given that you
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don’t know anything about either one -- or
even what the higher and lower values are
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inside them -- one is as good as the other.
But the math in favor of switching is… kinda
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compelling.
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Let’s say that the envelope you choose contains
X dollars. We don’t know how much X is,
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just that there’s something in the envelope
we can call X. The probability of that envelope
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containing the smaller value is ½, and the
probability of it containing the larger amount
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is ½. It's 50%.
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The other envelope either contains twice that
amount -- 2X -- or half that amount, 1/2X.
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If X turns out to be the higher value, then
the other envelope contains 1/2X. If your
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envelope's X is the lower value, then the
other envelope contains 2X. Make sense?
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I’ve talked about expected value in a few
other Vsauce2 videos. It’s the average of
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the possible outcomes of a series of probabilistic
events, which we can use to identify the most
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advantageous course of action. So, to find
the expected value of the envelope we didn’t
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choose, we add the value of the only two possibilities
weighted by their probability. Like this:
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½ (2X) + ½ (1/2X) = 5/4 X equals five over
four X. Half of the time the other envelope
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will contain 2X, and half of the time the
other envelope will contain one-half X…
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which equals an expected value of… five-fourths
X.
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5/4 X is 25% more than just X, so our best
possible choice is to switch to the other
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envelope.
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Cool. We did it. BUT THAT MAKES NO SENSE.
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You have a simple, clean 50/50 chance of choosing
the envelope with more money inside. Switching
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can’t change that... but the math just told
us that it can.
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Don’t worry, though. It gets weirder. What
if I said you could switch envelopes again
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if you wanted to? Well, then you’d just
go through the same process over and over
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again, always switching, never opening, forever
and ever and ever and erver and nerver and
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blerver -- those aren't words but you get
the point. Because switching makes the most
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mathematical sense, with switching gaining
you a 25% advantage. And I just proved it.
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Sooooooo.... Let’s come back to math later.
It’s time for logic.
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Logician Raymond Smullyan poses two scenarios
for the paradox: First, the risk of gaining
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twice the amount in your envelope is worthwhile
when the alternative is only losing half of
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it. You’re gaining X or you’re only losing
half of X. The potential payoff is twice as
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high as your risk of loss. Makes perfect sense.
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But one envelope contains X, and the other
contains 2X… so you’re either gaining
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X or losing X. Also makes sense.
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But both can’t be true. So we’re going
to do this problem again using economics professor
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Barry Nalebuff’s variation.
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Let’s say that we give an envelope to Player
1, and then flip a coin. If it’s heads,
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we put double that amount in the second envelope
and give it to Player 2. If it’s tails,
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we put half that amount in the envelope and
give it to Player 2. Neither player knows
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how much is in their envelope or the outcome
of the coin toss. Once they open their envelopes
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in secret, they can switch with each other
if they both agree to switching.
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Player 1 opens their envelope and finds $10.
With equal probability, they think that switching
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to Player 2’s envelope will get them $20
(for a $10 gain) or take them down to $5 (for
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a $5 loss). Player 1 decides to switch because
switching gives them a greater possible reward
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relative to what they might lose.
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Player 2, who’s in another room, opens their
envelope and finds $5. They think there’s
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a 50% chance that switching to Player 1’s
envelope will get them $10 (for a $5 gain)
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or get them only $2.50 (for a $2.50 loss).
So… Player 2 also thinks that what they
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stand to gain is greater than what they might
lose. They are both absolutely sure that their
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upside is better than their downside.
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But it’s obvious that BOTH players can’t
have an advantage in this game. That’s impossible!
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If you’re confused at this point… good.
Martin Gardner, one of history’s greatest
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puzzle-solvers and paradox-explainers, knew
the answer but admitted that there just wasn’t
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a simple way to communicate the flaw in mathematical
reasoning.
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And sadly, that’s where our journey ends…
NO IT’S NOT. Because we CAN figure this
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out.
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Think of it this way: the total sum of money
in the envelopes is three units -- a small
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unit, and then a larger value that’s made
up of two smalls -- that combine to form a
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total of X. If you’ve got the smaller envelope,
its value is x/3, and if you have the larger
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envelope, its value is 2X/3. When you switch
from small to large, you gain x/3, and when
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you switch from large to small you lose x/3…
So, the expected value of switching is: ½
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* [(2X - X)/3] + ½ [(X - 2X)/3], which equals…
0.
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There’s just no value in switching. It doesn’t
help us and it doesn’t hurt us -- which
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is probably what you knew until we got math
involved. The Two Envelopes problem is a lot
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more than just X and 2X -- which is what makes
it a falsidical paradox.
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Because there appears to be a gain from switching
when we fail to consider that the states of
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the two envelopes don’t actually reflect
simple values of X and 2X. A 2X envelope is
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only 2X when it’s the larger amount, and
the ½ X envelope is only ½ X when it’s
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the smaller amount. Without recognizing that
difference, basic algebra playing off our
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common sense fools us into thinking that each
switch gives you a gain of 5/4, or 25%.
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It’s really a matter of perspective. Depending
on which envelope you’re holding, which
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one you think you have, which one you think
you don’t have, and which one you want to
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wind up with, your course of action might
be incredibly clear… or not. And then clear
00:11:09.680 --> 00:11:13.420
again. And then not again.
00:11:13.420 --> 00:11:19.910
Until you realize that no matter if you diagnose
it mathematically, no matter if you deconstruct
00:11:19.910 --> 00:11:28.000
it logically, no matter what you do, when
it comes to the two envelope paradox, you
00:11:28.000 --> 00:11:32.220
were right all along.
00:11:32.220 --> 00:11:42.779
And as always, thanks for watching.
00:11:42.779 --> 00:11:51.060
Hey! I actually started developing this video
and started shooting it when Brilliant.org
00:11:51.060 --> 00:11:56.839
came to me about sponsoring it. I really like
working with Brilliant because it's an entire
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platform based on learning math and science
by playing with them. Something you may have
00:12:03.110 --> 00:12:09.990
noticed -- I’m kind of a fan of. And surprise
surprise, they have a really good explanation
00:12:09.990 --> 00:12:16.550
of these Perplexing Probability scenarios!
Their interactive puzzles help you expose
00:12:16.550 --> 00:12:21.370
misconceptions, and you’re learning by playing!
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This is just one of the 60+ courses on Brilliant
that teach you by walking you through puzzles
00:12:27.930 --> 00:12:35.800
and guiding you in figuring out the solutions.
Basically teaching you how to think. Which
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I would highly recommend. So head over to
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Go check it out. It’s awesome. Okay. Bye.