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Number Theory/Hypothetical Game

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Question
We play a hypothetical game.  In round 1, I think of any 10-digit number and you try to guess that number.  You are either correct in guessing that number or you are incorrect.  In round 2, I think of any 10-digit number again and you try to guess that new number.  You are either correct in guessing that new number or you are incorrect.  We play an infinite number of rounds.  Because we play an infinite number of rounds, I suggest that you are incorrect an infinite number of times, however, you are also correct an infinite number of times.  In conclusion, you would be correct as many times as you would be incorrect.

The basis of my conclusion lies in the transitive property (although there may be other ways to illustrate it).  If the number of correct answers are the same as the number of natural numbers and the number of incorrect answers are also the same as the number of natural numbers, then the number of incorrect answers and the number of correct answers are the same amount of elements.  I can say that you would be correct 50% of the time.

Am I correct in my analysis?  If not, please share with me where I made any errors.

Thanks,
Jimmy

Answer
Assuming you guess starting with 0 000 000 000 and end with 9 999 999 999, there are 10 billion possibilities.  Only 1 of these is correct.

I can then split this table of guesses into two tables.  The table of good ones will have only one right answer for every 9,999,999,999 answer is the other table.

If the two tables are chosen at random, there is a 50% chance of choosing the right table.
However, there is only 1 in 9,999,999,999.

When comparisons are made between two sets of infinity, a comparison like this is not looked at in this way since there aren't an infinite number of possibilities.

It could be said that this follows the law of large numbers.  Consider a situation where you were allowed to choose a digit randomly each time.  Even though there are an infinite number of choices, roughly one tenth will be of each number in the set.  This can then be extended to 10 billion choices and the same conclusions be drawn.

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