Monty Hall: Don’t Switch Doors, It’s a Trap
This Game Is Not Right: The Beginning
When I realized that, no matter which door I picked, it was always mathematically disadvantaged, I felt trapped. It immediately reminded me of a quantum experiment where an electron changes behavior as soon as it’s observed. I thought, “This can’t be!” It was as if a calculation was relentlessly working against my chosen door. I turned the problem inside out, and what I discovered is fascinating. Being forced to pick the BC block felt like someone saying: in an equilateral triangle ABC, side BC is the most important! But that made no sense—I couldn’t let that slide.
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A Reminder: The Classic Reasoning
A quick reminder of the classic rules: three doors, one hides a car, the other two goats. You pick door A. The host knows where the car is and must open another door with a goat, say C.
Initial probability of A: P(A) = 1/3
Probability of the BC block: P(BC) = 2/3
By opening C, the host conveys critical information: A stays at P(A) = 1/3, and B rises to P(B) = 2/3. This is the standard conditional probability. The usual conclusion: switch doors.
First Troubling Thoughts
The importance of the host’s action haunted me. His move is predictable, sometimes constrained, and one might think it conveys no useful information. Since we know in advance that he will eliminate a goat, it could seem merely a way to prolong the show. Perhaps the real choice begins only when two doors remain: A and B, and the 50/50 intuition kicks in.
Other examples to visualize:
Battle Royale: three finalists (1/3 each), one is eliminated randomly or by design. Two remain → chance for each: 1/2.
Who Wants to Be a Millionaire: three remaining answers, the host eliminates a known wrong one after a lifeline. This conveys no info about the two left → 50/50 intuition.
Variant with two doors A and B: the host opens B and reveals two new doors B’ and B’’. He opens B’’ (goat eliminated) → A and B’ remain at 1/2. Nothing more than the ABC game in disguise; labels distort perception and change results?!
These small victories against uncertainty cracked the dogma that you must always switch. Yet some emerged from cases where conditional probability didn’t fully apply—offering glimpses of balance, but never the full picture.. it wasn’t enough to explain everything. Tension rose, the mind wavered, each choice seemed to reveal a deeper secret, and curiosity refused to calm.
The Alternative Approach: The Real Twist
At this point, confusion was total. Conditional probability calculations are correct; the 50/50 intuition makes sense in some variants. But is there an angle that favors my door?
Let’s revisit: P(A) = P(B) = P(C) = 1/3, so P(BC) = 2/3. But wait… blocks AB or AC also equal 2/3. What if we chose AB or AC as the reference block, one that includes my door instead? As far as I know, there’s no rule preventing us from taking one of these blocks as the reference, one that includes my door.
Choosing A → block AB = 2/3 → host opens B → all probability transfers to A → don’t switch
Choosing A → block AC = 2/3 → host opens C → same result → don’t switch
Everything becomes dizzying: that’s it! The classic approach is biased, even directed. Focusing only on BC ignores the other blocks that also weigh 2/3. The outcome depends on the initial framing. “Always switch” is therefore not an absolute truth, but the consequence of a particular model. Depending on which block you apply the conditional update to, A can be favored—or not. Like a magician making you look right while everything happens on the left. This is my alternative approach, which I find more complete and honest! It looks at the problem from all possible angles!
Reflections
The real lesson of Monty Hall? Logic and probabilities can be manipulated by the chosen frame. What seems obvious can hide a subtle, almost teasing trap. In complex problems—finance, economics, trading, scientific studies, sports betting, poker—you shouldn’t blindly trust imposed models or gurus promising miraculous probability hacks. The effectiveness of a model depends entirely on its assumptions and on how you conceptualize the problem. Sometimes, staying humble in the face of uncertainty is the best strategy. The more certain we feel, the greater the risk—and a wrong move could be devastating.