Top Puzzle 15: The Two Envelopes Paradox | Puzzles for Interview - Interview Preparation PDF Download

Introduction

In this puzzle, you are given two envelopes, each containing a positive integer dollar amount, and both amounts are different. You do not know the values of these amounts. You randomly select one of the two envelopes and look at the amount inside. You have the option to switch envelopes if you wish. A friend suggests a strategy that will increase your probability of ending up with the envelope that contains the larger amount.

The Strategy

The strategy suggested by the friend is to toss a coin repeatedly. Let X be equal to 1/2 plus the number of tosses required to obtain heads for the first time. If the amount in the envelope you selected is less than the value of X, then switch envelopes.

The Solution

The answer given in the solution manual claims that this strategy does indeed help increase the probability of getting the better envelope. The probability of getting the better envelope is given by p = 1/2 x 1/2 P(B), where B is the event that a < X < b, with a and b being the smaller and larger amount of dollars respectively.

The Problem with the Solution

However, this solution seems flawed. Tossing a coin repeatedly has nothing to do with the contents of the envelopes, and opening the first envelope also gives you no useful information about the ordering relation between a and b. Even if you forget the coin tossing, the probability of "winning" is still 1/2, swap or no swap.

Interpreting the Puzzle

The statement "The values of these two amounts are modeled as constants that are unknown" means that the two amounts are randomly and independently generated numbers. Therefore, the solution manual is likely wrong.

Conclusion

In conclusion, the suggested strategy does not actually increase the probability of getting the better envelope. It is important to carefully examine the problem and solution to ensure accuracy.