Two Envelopes Paradox Calculator

Understanding the Two Envelopes Paradox

The Two Envelopes Paradox is a fascinating problem in decision theory and probability that challenges our intuitive understanding of expected value. It's a mathematical paradox that has puzzled philosophers, mathematicians, and statisticians for decades.

The Basic Setup

The classic version of the paradox is as follows:

1. You are presented with two sealed envelopes.
2. You're told that one envelope contains twice as much money as the other.
3. You randomly select and open one envelope, finding some amount of money (call it X).
4. You're then given the option to switch to the other envelope.

Should you switch? This seemingly simple question leads to a paradoxical conclusion when analyzed using expected value calculations.

The Paradoxical Reasoning

Here's the reasoning that leads to the paradox:

• After opening the first envelope, you know it contains amount X.
• The other envelope must contain either 2X or X/2 (since one envelope has twice the amount of the other).
• Both possibilities are equally likely, with a probability of 1/2 each.
• Therefore, the expected value of switching is:
  $$E(switching) = (1/2)(2X) + (1/2)(X/2) = X + X/4 = 5X/4$$
• Since 5X/4 > X, it seems you should always switch to maximize your expected value.

The paradox arises because this reasoning applies regardless of the amount X you find. Even if you switched and opened the second envelope, the same logic would suggest switching back to the first envelope. This creates an infinite loop of switching, which can't be optimal.

Resolving the Paradox

The resolution lies in understanding several key points:

1. Incorrect Probability Model: The paradoxical reasoning assumes that after opening an envelope, the other envelope is equally likely to contain twice or half the amount. This ignores the constraints of the initial setup.
2. Conditional Probability: Once you observe amount X, the conditional probabilities change. The proper analysis requires considering the prior distribution of possible amounts.
3. Bounded Distributions: In any real-world scenario, the amounts of money must come from some bounded distribution. With a proper prior distribution, the paradox disappears.
4. Symmetry Argument: By symmetry, before opening any envelope, the expected value of each envelope is the same. Opening one envelope and observing its contents doesn't change this symmetry for a truly random initial choice.

Mathematical Analysis

A more rigorous approach involves specifying a prior distribution for the amounts. Let's say the smaller amount follows a distribution with probability density function f(y). Then:

• The larger amount will be 2y with the same probability density.
• If you observe amount X, it could either be the smaller amount (in which case the other envelope contains 2X) or the larger amount (in which case the other envelope contains X/2).
• The conditional probabilities depend on the density function f(y).

With a well-defined prior distribution, the expected value calculations work out correctly, and the paradox is resolved. The simulation in our calculator demonstrates this by using specific amounts and showing that, on average, there's no advantage to switching.

Applications and Related Concepts

The Two Envelopes Paradox highlights important concepts in decision theory, probability, and statistical inference. It's related to other paradoxes like the Monty Hall problem and St. Petersburg paradox. Understanding these paradoxes helps develop better intuition for probability and expected value calculations in fields ranging from economics and finance to quantum physics and artificial intelligence.