Had the guest

We'll label the 3 prizes as G1 and G2 (goats) and C (car). There are 6 different arrangements:

# Door1 Door2 Door3

1 C G1 G2

2 C G2 G1

3 G1 C G2

4 G1 G2 C

5 G2 C G1

6 G2 G1 C

Pick a door, any door. Let's say Door 2. In 2 of the cases (3 and 5), you don't want to switch. In the other 4 case (1, 2, 4, and 6), you do want to switch. So using this logic, you want to switch 2/3 of the time so it makes sense to switch.

Let's look at it from Monty Hall's point of view. He knows where the car is. Let's say the arrangement is #5 from above. Now there 4 different possibilities:

1/3 of the time the contestant will pick door 1, Monty will show what's in Door3, and the contestant would want to switch.

1/6 of the time the contestant will pick Door 2 AND Monty will show what's in Door 1. The contestent should keep the original pick.

1/6 of the time the contestent will pick Door 2 AND Monty will show what's in Door 3. The contestent should keep the original pick.

1/3 of the time, the contestant will pick Door 3, Monty will show what's behind Door 1 and the contestent would want to switch.

So, again, 2/3 of the time the contestent should switch.

I guess I convinced myself that it's better to switch. I would prefer that someone find a hole in my logic!

]]>Yes, it is better to switch. When you picked the door, there were 3 to choose from and each had an equal chance. So the chance of you picking the right one was 1/3, meaning that the chance of you being wrong was 2/3.

Then the host removes an option, so now there are only 2 doors. We've established already that your door has a 1/3 chance of being right, so that means that switching must give you the 2/3 chance.

]]>**Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?**

Whaddaya reckon?

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