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Number Theory/Infinite set question

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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 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.  

Thank you,
Jimmy

ANSWER: Hi Jimmy
Of course you are.  The probability of success is 1/2.  So you don't need to bring infinity into it.
Best wishes
vijilant

---------- FOLLOW-UP ----------

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.

Thanks,
Jimmy

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.

Regards

vijilant

---------- FOLLOW-UP ----------

QUESTION: Vijilant,

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.

Answer
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.

Regards

vijilant

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Vijilant

Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

Experience

Teacher of math for 53 years

Organizations
AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

Publications
Journal of mathematics and its applications M500 magazine

Education/Credentials
BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

Awards and Honors
State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 50 prize First class honours in OU BA Mathematics

Past/Present Clients
I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.

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