Paradox #11 (the big one): *The Two Envelopes Paradox* This is a genuine paradox generating ongoing debate among mathematicians, economists, and logicians. It drives new traders crazy since it undermines the basic Bayesian math we use to identify good trades. /1

2/ the Two Envelopes Paradox: I show you a blue envelope and a red envelope and tell you that one envelope contains double the money inside of the other (you don't know which has more money). I hand you the blue envelope and ask if you want to switch it for the red.

3/ at this point it seems obvious that the answer is no. You have exactly the same information about the blue and red envelope so why would you switch? If you do switch from blue to red, I could then ask again if you want to switch, and the same logic would apply.

4/ so what's the problem? Imagine that you open the blue envelope and see $10. Now you know that the red envelope must contain either $5 or $20. Switching now seems to be an obvious win. If you switch you're either making $10, or losing $5. EV = 0.5*$10 - 0.5*5 = +$2.50

5/ yet this will *always* be true, no matter what you see in the blue envelope. If you had switched to the red envelope before opening either of them, and then opened the red envelope, this would suggest that you would want to switch again after opening.

6/ A slight tangent on how dangerous expressing the unknown quantities can be in variable form. If we define the blue envelope has having quantity A, then switching means either gaining A or losing A/2, switching EV+. But if we define the envelopes as having X and 2X,

7/ then switching means either gaining X or losing X, for EV neutral, so no point in switching. Both ways of expressing the relative cash amounts in the envelopes seem correct at face value but lead to different conclusions.

8/ I called this a tangent, because while it touches the problems that give rise to the paradox, it doesn't resolve the paradox. The paradox is really about what happens after you open one envelope.

9/ This paradox lacks clear resolution, but the most compelling explanation for me is the following: if blue envelope cash = A, then in asserting that the value of the red envelope = 0.5*2A + 0.5*(A/2), we're using a single variable "A", for two scenarios, only one can be true.

10/ that what we should be doing in considering two discrete scenarios, one in which the red envelope has more money, one in which the blue envelope has more money. Value of red envelope = 1/2* ( (EV of red given red is > blue) + (EV of red given red is < blue)).

11/ this reduces to the expected value of the red envelope = 1/2 (x + 2x) = EV of blue envelope. Extending this to opening an envelope - let's say that the two envelopes contain $10 and $20. Switching from one to the other is either +$10 or -$10, no asymmetry.

12/ In plain language, there's real two different possible scenarios, and only one can actually happen. Either the amounts in the envelope are (A, 2A), or (A, A/2), and we don't know which one. EV = 1/2 * (A/ 3A/2) + 1/2* (-A/2 / 3A/4) = 0

13/ this gives the "correct" conclusion after opening an envelope. We open the blue envelope and see $10 and plug that in as "A". We now get EV = 0 to switching.

14/ at its core, I think this paradox arises from conflating two different types of probability. When we toss a fair coin, we can say with 100% confidence and precision that it is *exactly* 50% to land on heads and 50% to land on tails. We have perfect knowledge,

15/ the probability is intrinsic to the process. In contrast, consider the scenario where I flip a coin but don't show you the result. The coin flip has already happened, it's either heads or tails. You would correctly reason that without any other info, it's 50% to be heads.

16/ but here the probability references not the state of the coin, but rather your own knowledge of the event. If all of this still leaves you a bit unsatisfied and confused...you're not alone.

17/ specific takeaways: have to be careful using probabilistic math in decision making and test for logical edge cases and contradictions. The "simple" formulas don't always lead to good decisions.

18/ general takeaway: there's really two different types of probability - a probabilistic process (like a coinflip), and a probability estimate that arises from our lack of knowledge. Conflating the two leads to contradictions.

19/ of course the smart solution: set both envelopes on fire and buy BTC since fiat is worthless.