[JeffJor]

Mr. Bertrand has (exactly - this needs to be included) two children (not twins, which is not quite the same as different ages). A gender, and a day of the week, that apply to at least one of his children have been written inside a sealed envelope. What is the probability that both children have that gender?

In this problem, we have no gender- or day-specific information. So the answer can only be the probability that he has two of the same gender. Which is 1/2.

Now open the envelope. If the answer changes to P based on what you see written, it has to change to the same P regardless of what you see written. Which means you didn't need to unseal the envelope; the answer was P before, not 1/2.

This is what Joseph Bertrand identified as his Box Paradox in 1889. That word was used to describe an actual contradiction, not a non-intuitive result. It disproves any answer except P=1/2. FOR ANY OF THESE PROBLEMS.

In fact, it is the same reason why the Monty Hall Problem's answer is what it is. Many "explanations" will claim that your original probability can't change, but never justify it. This is the justification - if it changes when one door is opened, it must change the same way when either door is opened.