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Let us suppose at present

1 dollar = 1 euro

After 1 year

There is 50% chance that 1 dollar = .80 euro ...[1]

And there is 50 % chance that 1 dollar = 1.25 euro ...[2]

Therefore expected value of 1 dollar after 1 year = .5*.8+.5*1.25 = 1.025 euro ...[a]

But statement 1 and 2 can be written as

There is 50% chance that 1 euro = 1.25 dollar ...[3]

And there is 50 % chance that 1 euro = .8 dollar ...[4]

Therefore expected value of 1 euro after 1 year = .5*.8+.5*1.25 = 1.025 dollar ...[b]

But statement [a] and [b] cannot be both right.

What is wrong here?

I don't know if this fits here? But this riddle was posed in this stat lecture https://youtu.be/UZjlBQbV1KU?list=PL2SOU6wwxB0uwwH80KTQ6ht66KWxbzTIo

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  • $\begingroup$ This is closely related to the two envelopes paradox: en.wikipedia.org/wiki/Two_envelopes_problem $\endgroup$
    – Aaron Montgomery
    Commented Jan 23, 2019 at 15:52
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    $\begingroup$ This is known as Siegel's Paradox . $\endgroup$
    – lulu
    Commented Jan 23, 2019 at 16:49
  • $\begingroup$ @lulu Sir, so if I understand correctly. In this case since expected value of both is same in terms of percentage change. The advisable strategy is to not buy the other currency(dollar or euro) and hold what you have? $\endgroup$
    – q126y
    Commented Jan 24, 2019 at 7:11
  • $\begingroup$ It's more subtle. My understanding (a bit stale) is that Siegel profits which are attributable to stochastic inflation are illusions (you get more dollars in worlds in which dollars are worth less, and fewer dollars in worlds in which dollars are worth more). However, if Siegel profits are attributable to actual changes in relative purchasing power then they are real. Here is a more detailed article on the subject. $\endgroup$
    – lulu
    Commented Jan 24, 2019 at 10:58

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