You’re given two indistinguishable envelopes, each containing money.
One envelope contains twice as much money as the other. You may pick one envelope and keep whatever amount it contains.
However, after picking, but before opening the envelope, you’re given the chance to switch your envelope with the other one.
The question is: Should you switch?
The paradox is based on a seemingly logical argument that indicates you should always switch:
- Suppose the amount in your chosen envelope is “x” dollars.
- The other envelope could then contain either 0.5x dollars (if you’ve chosen the one with more money) or 2x dollars (if you’ve chosen the one with less money).
- Thus, there’s a 50% chance the other envelope has 0.5x dollars and a 50% chance it has 2x dollars.
- The expected value of switching, based on this reasoning, is 0.5 * 0.5x + 0.5 * 2x = 1.25x. Since 1.25x is greater than x, it seems like you should always switch.
According to the above logic, it seems you should always switch since, on average, you stand to gain 25% more money by switching. But this contradicts our intuition, as both envelopes were indistinguishable at the start, and there shouldn’t be any advantage in switching.
So where does the paradox lie?
The Two Envelopes Paradox is a classic example of a problem where the maths seems sound, but the conclusion feels counterintuitive.
Here’s the resolution to the paradox:
The flaw lies in the assumption about the possible values in the other envelope based on the value in the envelope you initially pick.
To explain further:
- The Flawed Reasoning: When you pick an envelope and assign the value “X” to its contents without knowing X, you’re essentially treating X as a random variable. However, the analysis that follows makes assumptions about the distribution of possible values of X, which may not be valid.
For instance, let’s assume you pick an envelope, and it has $10. According to the original reasoning, the other envelope could have either $5 or $20 with equal probability. But if you had instead initially picked the other envelope and found it had $5, the reasoning would then say the first envelope could have either $2.50 or $10. So, depending on the arbitrary choice of the first envelope, the problem defines different sets of possible values for the contents of the envelopes, which isn’t consistent.
- Another Perspective: Let’s consider a concrete scenario to highlight the flaw. Imagine there’s a 50% chance the smaller envelope has $5 and a 50% chance it has $100. This means the larger envelope could have either $10 or $200. If you pick an envelope and see $100, you might be tempted by the flawed reasoning to think there’s a 50% chance the other envelope has $50 and a 50% chance it has $200, giving an expected value of $125 for switching. However, in reality, if you see $100, the other envelope can only have $50. The original “expected value” argument fails because it doesn’t account for the known distribution of amounts.
- Conclusion: In essence, the paradox arises from a subtle misapplication of probability and expected value. The distributions of possible amounts in the envelopes are not established or consistent, so an expected value calculation based on the assumption of “X” is not valid.
Given that you know nothing about the distribution of the amounts in the envelopes, there is no advantage to switching. The expected gain from switching is the same as the expected gain from not switching, so the choice doesn’t matter.
