Your problem is whether I am required to play the same door each time. If I am, then you are right. If I am not, my expectation value is 3 cards if I play 10 times, or 10 cars if I play 30 times.

Actually, it doesn't matter or not if you are required to play the same door each time, and empirical reality supports my viewpoint. I did that to try to simplify the example. But obviously you can see from this example that if that is the strategy you chose, then it would not be 50:50, correct? You have just admitted that right there. So, how can you say that it is 50:50 when in fact I can cite this strategy which is not 50:50? A contradiction, don't you think?

Let me run an empirical experiment for you, our own Monty game. I have 10 cards here. One is a Joker. I will shuffle, pick a card at random. Then, I will prune the non-picked cards down to only one card, but only throwing out non-Jokers as I go.

Your hypothesis suggests I should have a 50:50 chance of picking the Joker on the first guess. Mine is that I only have a 1/10 chance and therefore you should always elect to switch. Let's play!

Monty Trials:

1. Non-Joker. I should have switched.

2. Non-Joker. I should have switched.

3. Non-Joker. I should have switched.

4. Non-Joker. I should have switched.

5. Non-Joker. I should have switched.

6. Non-Joker. I should have switched.

7. Non-Joker. I should have switched.

8. Non-Joker. I should have switched.

9. Non-Joker. I should have switched.

10. Non-Joker. I should have switched.

Ok, that was quick to do. If it is supposed to be 50:50, how do you explain the above results? Try it yourself Doc. It took me about 2 minutes.

You have a hypothesis which does not match up with reality. But you are clinging to that hypothesis. Does a good scientist stick to a hypothesis if it does not match up with reality, or do they modify their hypothesis?

If you think I am mistaken and that my results were highly unusual, do the test. In fact, anyone else who is reading this, I encourage them to do the test too. It is extremely quick to do.

Naturally, with a larger sample size, the answer will converge on 1/10th chance of picking the Joker.

Let us have the flexibility to undo lodged beliefs so we can see things as they really are. This Monty problem is counter-intuitive, so I can understand why it is hard to see.