An unfair game
You write down 2 distinct real numbers from [0,1] and seal them in envelopes. I pick one envelope, open it, and guess whether the number that I see is greater or less than the number in the other envelope.
I claim that I have a winning strategy, i.e. a strategy that allows me to guess correctly with probability greater than 50%.
Such strategy does exist.
It is important to note that you don’t pick numbers “at random”, you pick them “at will”. That is, I cannot assume that they obey some certain distribution. I intend to win no matter how you try to counter my strategy.
I pick a random number from (0, 1). If it’s greater than the number that I see, I say that the other number is greater, otherwise I say that the other number is less. My guess will be correct in more than 50% of cases.
Let the hidden numbers be a and b, a < b. And let my generated random number be x.
I pick a with probability 1/2. If x > a then I win. The probability of this is 1-a.
Or I pick b with probability 1/2. If x < b then I win. The probability of this is b.
The total probability of me winning is 1/2*(1-a) + 1/2*b = 1/2 + (b-a)/2 > 1/2
Obviously, the player who picks the numbers can make the loss arbitrarily small by picking very close numbers.
How does this work?
Imagine that you know some number between a and b. Then your win is guaranteed. Picking a random number ensures that sometimes (with some positive probability) it will fall in between a and b. That makes your chance greater than 1/2. Note that the distribution of x doesn’t have to be uniform. It just needs to be positive everywhere. So that for any interval the probability of x falling in it is positive.
Do we extract information from the random numbers?!
No. But randomness prevents our opponent from preventing our winning 😉 Imagine that the opponent can predict our random numbers. Then we won’t be able to win. And in the finite case with a deterministic random generator, the opponent will sooner or later learn to predict it. (See https://bbzippo.wordpress.com/2010/05/23/hunting-problem/ )