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So suppose one day you're taking a stroll and a strange guy comes up and says "Hey here's
an envelope, take it, it's yours!". So you happily take the envelope, and find there's
money inside. Just then he whips out another envelope. He tells you that in this
envelope there is either half, or double the amount of money in your envelope. Would you
like to swap envelopes?
Recalling the statistics you've learnt, you quickly compute the expected value of the
second envelope, and realise that it is one and a quarter the value of the first, and
you therefore decide to make the swap.
The next day, you're taking a stroll again, and the same guy comes along. This time he
has two envelopes. He says to you "These two envelopes contain money, but one has twice
the amount of the other! Go ahead, pick one." Since you don't know which has more money,
you take one at random. Now he asks you if you want to swap envelopes. You realise that
since the first envelope was picked at random, the other could contain either half or twice
the amount of money, with equal probability. Recalling the argument from the previous day,
you decide to make the swap. However, now you realise that the envelope he's holding,
which was the one you originally chose, contains either half or twice the amount of money as
the one you're holding now. So you swap again. And again, and again, and again. But no matter
which envelope you're holding, the other seems to be the better envelope. So what's really
going on? Which envelope is really better? This is the two envelopes problem, a famous
paradox in statistics.
The reason we ran into this paradox is because we made a very poor assumption. We assumed
that regardless of how much money the envelope you're holding contains, the other is equally
likely to contain half or double the amount. Let's see what's wrong with this assumption.
However be warned, the rest of this video contains some serious mathematics.
Since you chose the envelope at random, you're equally likely to choose the one with more
or less money. Doesn't that mean that the other envelope will be equally likely to contain
half or twice the amount? The answer is no, it depends on how much money is in your envelope.
The probability of the other envelope containing twice the amount as yours, conditioned on
the amount in your envelope, is not half. Suppose the amount of money in the envelopes
are drawn from a certain probability distribution. We find that if the envelope we have contains
a large amount, the other is likely to contain less. Similarly, if we find that our envelope
contains a small amount, the other is likely to contain more. Is it possible that the money
is distributed in such a way that regardless the amount in your envelope, the other is
equally likely to contain half or double the amount? Let's try to construct such a distribution.
Let the amount of money in the first and second envelopes be X and Y respectively, both being
random variables. We assume that regardless what the value of X is, Y will be equally
likely to be equal to half or twice that value, and vice versa.
This gave us the result that each envelope is expected to contain 1.25 times the amount
in the other, which seems really troubling.
Let's see what the probability distribution of money in the envelopes look like. Since
both envelopes are similar, X and Y are identically distributed. For simplicity, we'll let X and
Y take discrete values, with probability mass function f(a).
Suppose one of the possible outcomes is one dollar. If I pick an envelope containing one
dollar, the other must contain two or half with equal probability. In that case, two
and half must be possible outcomes too. If we follow this reasoning, we see that four,
a quarter, and all other integer powers of two must be possible outcomes. Therefore,
there must be infinitely many possible outcomes. Furthermore, they must be equally likely.
Now let's ask the question: what is the probability that an envelope contains between a thousandth
of a dollar to a thousand dollars? That is 19 outcomes out of infinity, which is zero.
What is the probability that an envelope contains between a millionth of a dollar to a million
dollars? That is 39 possible outcomes out of infinity, which is still zero. In fact,
the probability of an envelope containing any finite nonzero amount, is zero. In other
words, the envelopes can only take values of zero or infinity. This is the result of
the earlier assumption that the second envelope is equally likely to contain either half or
twice the amount in the first envelope, regardless of the amount inside.
And the result from earlier, that each envelope is expected to contain 1.25 times the amount
in the other, is indeed possible when both are zero, or both are infinity.
However, since we require our envelope to actually contain some money, and money has
to be finite, such a distribution is not allowed, and hence the assumption cannot be true.
In summary, in the first scenario where we know that the second envelope is equally likely
to contain half of double the amount in the first, swapping indeed gives a higher expected
return. However in the second scenario, these probabilities are not equally likely and depend
on the distribution the money is drawn from, and the amount in your envelope. Without access
to this knowledge, there is no difference in swapping or staying.
Thanks for watching, let me know in the comments what you think and what else you would like
to see. There will be more to come, so remember to like and subscribe. I hope you got something
out of this, take care.