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Imagine that two big urns are put in front of you, and you know that one of them contains ten balls and the other a million, but you are ignorant as to which is which. You know the balls in each urn are numbered 1, 2, 3, 4 ... etc. Now you take a ball at random from the left urn, and it is number 7. Clearly, this is a strong indication that that urn contains only ten balls. [...]
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This isn't true. The fact that you chose ball number 7 doesn't say anything about which urn is which.
It is true that before you choose a ball, you are more likely to pick 7 if you pick from the urn with 10 balls. Reasoning the other way doesn't work, though.
Look at it this way: before you chose a ball, you had to choose an urn. At that point, you had a 50% chance of choosing the urn with 10 balls. Then you pull out ball #7. That gives you no new information about which urn you have selected! Sure, it is unlikely a priori that you would choose #7 from the million urn, but the fact is that you did pull out number 7. If you'd pulled out numbers 11-1000000, you'd have new information, but you did not. Your odds of having chosen the ten ball urn are still only 50%.