Number Theory/Infinite set question
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 wrong 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?
Please share your thoughts and ideas on this.
ANSWER: Hi Jimmy
Of course you are. The probability of success is 1/2. So you don't need to bring infinity into it.
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QUESTION: I am thinking that if it is true that I don't have to bring infinity into it, it's like saying the probability of winning the lottery would be 1/2. So, if we played 100 rounds, you should on average win 50 rounds? Clearly, I am missing something here. (You may be a good guesser, but I doubt you're that good!)
By the way, I tried to donate some money to you from the 'donate' link that takes me to paypal. When I try to enter my billing address for my Visa card, it only accepts countries in Europe (by a drop down menu). I live in the U.S. and it, therefore, won't let me donate.
ANSWER: Hello Jimmy
Clearly we each have the same chance of winning, so the probability of success is 1/2.
If we have played n games and you have won x, the experimental probability p of you winning is x/n. As x tends to infinity, p tends to 1/2.
Of course, the probability of winning the lottery is far,far less than 1/2.
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I sincerely apologize for the confusion in my original question. I feel bad that I did not phrase the rules of the game properly. In the game, we never swap sides. I am always the one who thinks of a new 10 digit number and you are the one who always try to guess it. Obviously, there is no object to this game, its just an exercise. At first, I was thinking that there was no way you could guess the number because it changes in each new round. Then I thought that if we played forever, you would eventually be correct. Then I thought that if you were correct once, what would prevent you from being correct again and again? After all, we are playing forever! If you were correct an infinite number of times, that would also be the number of times that you would be incorrect. So, you would be correct 50% of the time. My logic seems to be flawed, but I can't see where. Again, I apologize for the confusion in my original question.
Your number each time could be any of 9*10^9 different numbers. The probability of your friend being successful is then (1/9)*10^(-9).
You cannot think of infinity as a number. It isn't, and certainly doesn't obey the usual rules of algebra or arithmetic. For example: let x = infinity. Then x + x = x. Solve to get x = 0.
Or x^2 = 3x. x(x-3) = 0. x = 0 or 3.