Advanced Math/Hypothetical Game
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. I am always the one who thinks of a 10-digit number and you are always the one who tries to guess that number. I suggest that there is a very low probability of you ever guessing the correct number. But, because we play an infinite number of rounds, I suggest that you are correct an infinite number of times. The reason is that the probability of you winning a round is not zero. Therefore, if we played enough rounds, you will eventually guess correctly. If it is possible for you to guess correctly, you will eventually guess correctly again, and again... Is my assessment correct? What are your thoughts?
ANSWER: You are correct, the number of correct answers will be infinite.
Roughly speaking, the probability of correct answers is a set with non- zero measure
And is one- to-one with an infinite set (integers) -> infinite.
Let me know if this makes sense.
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QUESTION: The number of correct answers are a one-to-one correspondence with the set of integers. I suggest that the number of incorrect answers are also a one-to-one correspondence with the set of integers. Therefore, by the transitive property, the number of correct answers are a one-to-one correspondence with the number of incorrect answers. If that conclusion is correct, there should be the same number of incorrect answers as there are correct answers. So, it be said that you would be correct 50% of the time. It seems strange that I concluded this considering that the probability of getting a single correct answer is a very small number. If my analogy is correct, it would also be strange if the rules of the game were changed to guess a 1000-digit number instead of a 10-digit number. (You would also be correct 50% of the time under the new rules.) Please let me know if I am in error in my thinking.
You have to be careful here because you are dealing with infinity, i.e. 1/2 of infinity is still infinity. You are actually asking what the cardinality of the one-out-of-10 set is (where 1/10 is the probability made rigorous using the law of large numbers). Look at Cantor's proof that the cardinality of a well defined subset of the integers, say the odd numbers, is infinite, as are the integers, using a one-to-one mapping argument. See Lesbeque references for measure theory and probabilities.