Actually, you do. In order to generate a random number, you have to know the probability distribution. Therefore your original assertion was right, and my analysis was right. I didn't get it 100% last time. In the problem the probability distribution is the 'trivial' uniform, infinite distribution and with a mean of zero you can actually beat better than 5050 odds with only the ability to generate a random number  you don't actually have to do it. If you are randomly given a second number without knowledge of the distribution, the result tells you something about the distribution again giving you better than 5050 odds. I'm done!
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