# Envelope Paradox

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Imagine being presented with two envelopes. The first envelope contains anywhere between $5 to $20, and the other contains either half or double in value. Looking at the first envelope, you are given the option to keep it or switch to the second.

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Contributed by: Akshay Mishra and Aditya Ponukumati (June 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

It is advantageous to choose the envelope with the higher expected value. Assuming a uniform probability density function for the random $5 to $20 amount in the envelope, the expected value can be viewed as the arithmetic mean of all possible envelope values.

Assume the first envelope to enclose an amount , . As such, the second envelope either encloses an amoun or . The expected value of this envelope is .

Therefore, switching to the second envelope on average nets 25% more money.

## Permanent Citation