9873

Envelope Paradox

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.
This problem is known as the exchange paradox. The Demonstration simulates the effectiveness of two different strategies: keeping the first envelope or switching to the second. The red data points represent the strategy of switching to the other envelope, and the blue data points represent keeping the original envelope. The axis is the cumulative amount of money, and the axis is the number of trials.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+