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The probability of winning each round is not zero, it is 1/10^10.

Out of the 10^10 ten-digit numbers, one of them is the winner.

So the probability of losing each round is 1 - 1/10^10.

The probability that you guess N times and get it wrong every single time is (1-1/10^10)^N.

So the probability that you get it right even once is:

1-(1-1/10^10)^N

As N grows larger, this quantity approaches 1. This means that over time, as the number of guesses goes to infinity, the "guesser" will actually win the game

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