Explain why the same empty door can lead to different odds.
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First lesson in this subject: Count equally likely outcomes
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A prize is equally likely behind any of 3 doors. You choose door 1. An informed host always opens an unchosen empty door, choosing uniformly when two are available. You see door 3 opened empty.
That reveal has probability 1/2 if the prize is at 1, and 1 if it is at 2. Multiply each by the starting chance 1/3 to get the weights 1/6 and 1/3. Scale the weights so they add to 1: staying wins with 1/3, switching with 2/3.
You are reading the opening. Members read every lesson and guide in full. A free account reads the first lesson in each subject in full.
You are reading the opening. Your free account reads the first lesson in each subject in full. Members read every lesson and guide in full.
First lesson in this subject: Count equally likely outcomes
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Practice · Open the doors, inspect the host’s rule and test when switching really helps.
Book · Count outcomes, update chances with new information, and compare decisions through worked examples.
Previous lesson · Track the changing counts when an object is not replaced.
Next lesson · Subtract the cost and weight every net outcome by its probability.
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