The trick here – and there is a trick – is a subtle use of equivocation: You make a claim about one problem but then explain the claim in terms of another problem that is subtly yet significantly different.

In the first problem, each envelope can contain any number. In the second problem, however, you require that the distribution of numbers have a "continuous probability distribution with non-zero density everywhere." These problems are not the same.

To see why, imagine an infinite number line representing all numbers. If you pick any segment on this line, say that between 0 and 1, the portion of the line that the segment represents is zero. Hence the probability density function over the segment is also zero, which contradicts the assumption made in your analysis.

If we repeat your analysis, this time using the distribution of numbers for your originally stated problem, we find that the probability of picking a number in between your two numbers is zero, and thus knowing the first number provides no benefit. Our intuition turns out to be correct.

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