A Clarification of the Monty Hall Problem

A Clarification of the Monty Hall Problem
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I have seen one too many people fall into the trap of looking at the MH problem the wrong way. I would like to fix that.

Part One below is a concise explanation of the correct solution to the Monty Hall problem.
Then, there is a very common pit I see people fall into with MH, and Part Two addresses this.

If you are not familiar with the Monty Hall problem, please utilize Google at this time.

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Part One: The Correct Solution

There are two goats and one car, hidden behind doors. This means that your initial pick will be a goat 2/3 of the time. Simple.

As the host *always* removes a goat after your initial pick, 2/3 of the time the remaining door hides a car, since 2/3 of the time your initial pick was a goat.

The other third of the time, you picked the car initially, and switching gives you the goat.

But the *majority* of the time, 2/3, switching is the right choice. More often than not. So you should always switch. 2/3 of the time switching your pick will give you the car. Sticking with your initial pick would only give you the car 1/3 of the time.

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Part Two: The Common Pitfall

This is where I usually find people spluttering something to the effect of "But-- but-- you're still picking between two closed doors! It's still 50/50! You still don't know which is which! How has removing a door changed anything, how has it given you any extra information?? It's still between two unknown doors!!"

If you find yourself following reasoning that even remotely resembles the above exclamation, read on.

Let's imagine a different scenario. Let's imagine I present to you a jar of 999 red marbles and 1 blue marble, and with your eyes closed you reach in and take a marble. Which color are you most likely to have chosen, red or blue?

Before you open your eyes, I remove all other marbles from the jar EXCEPT one that is opposite to the color you chose initially.
So if you chose one of the 999 red marbles, all that's left is the red in your hand and the 1 blue.
If you chose the 1 blue marble, all that's left is the blue in your hand and 1 of the red marbles.

So if the goal is to get the blue marble, and with your eyes still closed, ask yourself: after I've removed all but one, should you switch your pick?

999 times out of 1000, you picked the red marble. The odds are 99.9% that you chose a red marble. Which means 99.9% of the time, you should switch so that you get the blue one.

I find that by carrying the numbers to a greater scale, the Monty Hall problem becomes clearer. You are *more likely to have picked a goat at first, because there are more goats than cars.*

And since the choice becomes binary after the host removes a goat -- 1 goat, 1 car -- the odds are 66% that you started with a goat, and only 33% that you started with the car, thus you're better off switching.

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I hope this has helped someone. Questions/comments to julianeden2@gmail.com.