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So A tells his number. Now the probability that A>B seems to jump to 0 because there are finitely many numbers smaller than A but infinitely many larger. This

by jeromec 16y ago

So A tells his number. Now the probability that A>B seems to jump to 0 because there are finitely many numbers smaller than A but infinitely many larger. This is not correct. You said B chose a positive integer. Therefore, there are not infinitely many larger numbers B's choice can be.

InclinedPlane 16y ago
Sure there are. No matter what finite positive integer you choose there are infinitely more finite positive integers greater than that number.
- jeromec 16y ago
  That's true, but what te_platt said was "two people (A and B) choose two positive integers at random" If two people, A and B, have chosen -- that's choose in the past tense -- two positive integers, there are not infinitely more positive integers B's number can be greater than A's number. Why? Again, because B has already made a choice.
  - InclinedPlane 16y ago
    Actually, it doesn't matter. If you rephrase the problem so that B picks after A and you ask the question "is B likely to pick a larger number than A" before B picks then the answer should be "yes". However, this is another example of being misled in reasoning. But not related to timing. The important revelation is understanding that there's no such thing as a method for generating truly random numbers across the entirety of the positive integers. Since there are infinitely many the probability of picking any one integer is 1/infinity. Any concrete random algorithm for positive integers will be hopelessly biased toward 0 (since it will almost certainly have a maximum value, and thus an infinite number of positive integers above that maximum). As long as both A and B use the same algorithm then the chances are equal that A or B will pick the larger number.
    - jeromec 16y ago
      If you rephrase the problem so that B picks after A and you ask the question "is B likely to pick a larger number than A" before B picks then the answer should be "yes". I agree, but that wasn't what was stated. You've "moved the goal posts" so to speak. ;) The OP said both A and B had chosen a number (but only one had revealed it). That introduces TWO finite values into the experiment. The timing for when B makes the choice makes all the difference.

by jeromec 16y ago

InclinedPlane 16y ago

Sure there are. No matter what finite positive integer you choose there are infinitely more finite positive integers greater than that number.

jeromec 16y ago
That's true, but what te_platt said was "two people (A and B) choose two positive integers at random" If two people, A and B, have chosen -- that's choose in the past tense -- two positive integers, there are not infinitely more positive integers B's number can be greater than A's number. Why? Again, because B has already made a choice.
- InclinedPlane 16y ago
  Actually, it doesn't matter. If you rephrase the problem so that B picks after A and you ask the question "is B likely to pick a larger number than A" before B picks then the answer should be "yes". However, this is another example of being misled in reasoning. But not related to timing. The important revelation is understanding that there's no such thing as a method for generating truly random numbers across the entirety of the positive integers. Since there are infinitely many the probability of picking any one integer is 1/infinity. Any concrete random algorithm for positive integers will be hopelessly biased toward 0 (since it will almost certainly have a maximum value, and thus an infinite number of positive integers above that maximum). As long as both A and B use the same algorithm then the chances are equal that A or B will pick the larger number.
  - jeromec 16y ago
    If you rephrase the problem so that B picks after A and you ask the question "is B likely to pick a larger number than A" before B picks then the answer should be "yes". I agree, but that wasn't what was stated. You've "moved the goal posts" so to speak. ;) The OP said both A and B had chosen a number (but only one had revealed it). That introduces TWO finite values into the experiment. The timing for when B makes the choice makes all the difference.