This page compares two famous probability puzzles that both ask if choices matter.
The goal of this page is present these problems, let you play with the simulators, and see the results for yourself.
The Envelope Problem
Two envelopes contain money. One has twice as much as the other. You choose one envelope, but before opening it you are allowed to switch to the other envelope.
Does it matter if you switch or keep the envelope you picked first?
Envelope Game
Pick an envelope. Before seeing either amount, keep it or switch. The higher amount is tracked as a win-like outcome, and payout is tracked as a multiple of the smaller amount.
Start a round, pick an envelope, then decide whether to keep it or switch.
Keep: higher envelope0 / 0
Switch: higher envelope0 / 0
Keep average payout0.00x
Switch average payout0.00x
Monty Hall
In the Monty Hall Problem, there are three doors hiding prizes. One hides a car (win) and the other two hide goats (lose). You choose a door, then the host opens a goat door from the two remaining doors you did not choose. You can keep your first door or switch to the only other unopened door.
Does it matter if you switch or keep the door you picked first?
Monty Hall Game
Pick a door. Monty opens a goat door you did not pick. Now keep your first door or switch to the remaining closed door.
Start a round and pick a door. Monty will reveal a goat, which is the whole trick.
Keep wins0 / 0
Switch wins0 / 0
Keep win rate0.0%
Switch win rate0.0%
Simulator and the Law of Large Numbers
Individual rounds can be noisy. A good strategy can lose one round, and a neutral strategy can look lucky for a while. The law of large numbers says that as independent trials pile up, the observed averages tend to settle near the true long-run rates.
The simulator runs three strategies for both puzzles: always keep, always switch, and randomly keep or switch with probability 0.5. In Monty Hall, switching should approach about two wins out of three, random should approach one win out of two, and staying should approach one win out of three. In the envelope game, all three should converge toward the same average payout.
Simulation Lab
Each simulated batch runs all three strategies on both puzzles so the long-run behavior can be compared directly.
Game
Keep / stay
Switch
Random
What changed?
Two Envelopes
0.00x avg
0.00x avg
0.00x avg
No data yet
Monty Hall
0.0% wins
0.0% wins
0.0% wins
No data yet
Monty Hall win rate
StaySwitchRandom
Envelope payout multiple
KeepSwitchRandom
Run a batch to draw the convergence lines.
Closing Remark
My bias is that the basic Two Envelopes problem is a fake paradox: it sounds like Monty Hall because it talks about switching, but it does not include the informed reveal that makes Monty Hall work. Monty changes the information state. The envelope game, at least in this version, just lets you trade one unknown for another.
