Envelope Dilemma

Solution & Explanation

The "Envelope Dilemma" is a classic probability problem that tests one's understanding of expected value, a fundamental concept in decision-making under uncertainty. Here's a detailed breakdown of the solution:

Problem Overview:

You are given two envelopes:

• Envelope 1: Contains a guaranteed $5,000.
• Envelope 2: Contains a 50% chance of holding either $10,000 or$1,000.

The task is to decide which envelope to choose based on expected value calculations.

Expected Value Calculation:

Expected value (EV) is a measure used in probability to determine the average outcome of a random event if it were repeated many times. It is calculated as the sum of all possible values each multiplied by their probability of occurrence.

1. Expected Value of Envelope 1

   • Since Envelope 1 contains a guaranteed $5,000, the expected value is straightforward:

     $E[X_1] = 1 \times 5000 = 5000$

   • Explanation: There is a 100% probability of receiving $5,000, so the expected value is simply$5,000.

2. Expected Value of Envelope 2

   • Envelope 2 offers a 50% chance of containing $10,000 and a 50$1,000. The expected value is calculated as follows:

     $E[X_2] = 0.5 \times 10000 + 0.5 \times 1000$ $E[X_2] = 5000 + 500 = 5500$

   • Explanation: The expected value computation involves multiplying each potential outcome ($10,000 and$1,000) by their respective probabilities (0.5 each) and summing the results. This gives an expected value of $5,500 for Envelope 2.

Decision Making:

• Comparison: The expected value of Envelope 2 is $5,500, which is higher than the expected value of Envelope 1, which is$5,000.
• Conclusion: Based on expected value alone, Envelope 2 is the better choice because it offers a higher average monetary outcome over repeated trials.

Additional Considerations:

• Risk Aversion: While the expected value favors Envelope 2, individual risk preferences might influence the decision. A risk-averse person may prefer the certainty of $5,000 over the variability of Envelope 2.
• Utility Function: If the decision-maker's utility function is nonlinear, the choice might differ. Utility functions that assign diminishing returns to monetary gains could make the guaranteed $5,000 more attractive.

In summary, while the expected value calculation points to selecting Envelope 2, the actual decision may vary depending on the decision-maker's risk tolerance and utility function. This problem illustrates the importance of understanding both statistical measures and personal preferences in decision-making under uncertainty.