The Two Envelopes Paradox: A mind-bending probability puzzle. Should you switch envelopes? Discover the surprising answer and challenge your intuition.
The Two Envelopes Paradox
You're given two envelopes. One contains an unknown amount of money, the other contains twice that amount. You pick one. Should you switch to the other envelope?
Problem Setup and Initial Choice
1. Two Envelopes, Unknown Amounts:
- You have two envelopes in front of you. One envelope contains an unknown amount of money, let’s call it X.
- The other envelope contains either twice that amount (2X) or half that amount (X/2).
2. Your Choice and Question:
- You pick one envelope but don’t open it. You’re asked if you’d like to switch to the other envelope.
- The question: Is it advantageous to switch? Would you switch?
Step-by-Step Logical Analysis
Step 1: Identifying the Initial Barrier
The key barrier in the problem is the temptation to think switching will increase your expected gain based on potential outcomes, which creates the paradox.
You reason that if your chosen envelope contains X:
- There’s a 50% chance the other envelope has 2X.
- There’s also a 50% chance the other envelope has (X/2).
Step 2: Asking the Key Question – Should You Switch?
To determine if switching is advantageous, let’s calculate the expected value if you switch.
Expected Value Calculation:
If you switch, the other envelope could contain either 2X or (X/2).
The expected value of switching would then be:
Expected value = [(1/2) * 2X] + [(1/2) * (X/2)] = 5X/4.
The expected value of 5X/4 being greater than the amount X you hold, you will always be tempted to switch.
The Paradox and the Barrier:
- This expectation might tempt you to keep switching back and forth indefinitely, as it seems switching will always increase your expected value.
- The paradox arises because this logic applies no matter which envelope you hold, making it seem like switching is always better, which is illogical and an unacceptable situation.
Step 3: Breaking Down the Paradox – The Core Realization
To overcome the paradox, you need to think differently.
Think for a moment that you are holding envelope A with equal probability of having X or 2X amount.
What if you held the envelope B? Would it have changed the situation?
Not at all. In that case also, the probability of your envelope containing X or 2X will remain same as 50%.
Clinching reasoning and breaking out of the tempting paradox is then:
There is no difference between two situations of holding envelope A or envelope B. If you switch from A to B, though the expected value calculation tries to trap you into an infinite loop, you know now that the probabilities of the envelope B containing X or 2X remain same as 50%, no different from the earlier situation.
Your decision is then: No switch. The two envelopes are equivalent, and there is no advantage in switching.
This is a good lesson on the truth of avoiding false temptation by thinking in a grounded manner, assessing what if scenarios and understanding the equivalence of the two situations.
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