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## Question

This is a result of a fallout from a previous puzzle.. You choose a point "randomly" on the positive side of origin on x-axis. This point is called A.

Choose an another point again on x-axis (again, positive). Call this point B.

What is the probability that B is closer to the origin than A? Is it 0? Or 0.5? Or something else?

If the question is not well-formed, please feel free to qualify it with "reasonable" statements.

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Depends on the technique of choice. If you choose first the point A and then B, the probability is zero, because A divides the axis on finite and infinite parts. (probability = N/infinity = 0)
If you will simultaneously "throw" these points on the axis, then the odds that B is to the left or to the right from A are equal. The answer is 50-50.
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Depends on the technique of choice. If you choose first the point A and then B, the probability is zero, because A divides the axis on finite and infinite parts. (probability = N/infinity = 0)
If you will simultaneously "throw" these points on the axis, then the odds that B is to the left or to the right from A are equal. The answer is 50-50.
How you like it?  So obviously (at least) one of the approaches is wrong (if not both). Which one and why?

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Asking about probabilities makes no sense unless there is a probability distribution.

Unfortunately there is no reasonable distribution on R+.

But an obvious distribution exists on interval [0,x], where x is any positive real number.

One thing we can do in such case is to try to solve the problem for [0,x] and then see

if the solution has a limit when we go with x to infinity.

With this approach it's obvious that the probability we look for equals 0.5 for every x,

so obviously it's limit is also 0.5 when x goes to infinity.

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Asking about probabilities makes no sense unless there is a probability distribution.

Unfortunately there is no reasonable distribution on R+.

But an obvious distribution exists on interval [0,x], where x is any positive real number.

One thing we can do in such case is to try to solve the problem for [0,x] and then see

if the solution has a limit when we go with x to infinity.

With this approach it's obvious that the probability we look for equals 0.5 for every x,

so obviously it's limit is also 0.5 when x goes to infinity.

For me, it's not obvious.
Imagine rain and two plots, one of which is twice larger than the other. Obviously, on a larger plot fall twice drops, i.e. probability for each drop to fall at a larger plot is approximately  0.66. And if the second plot is 1000 times larger, and if it is infinitely large?
So, for me it's not obvious. And for you?
Edited by koren

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Depends on the technique of choice. If you choose first the point A and then B, the probability is zero, because A divides the axis on finite and infinite parts. (probability = N/infinity = 0)

If you will simultaneously "throw" these points on the axis, then the odds that B is to the left or to the right from A are equal. The answer is 50-50.

How you like it? The order of choice, by symmetry, cannot matter.

The probability that A is finite is the same as the probability that B is finite.

Edit:

As witzar points out, the probability density function comes into play.

For a probability evenly distributed over an infinite domain, that function is zero for every finite region. That is to say the probability of choosing at random any finite number from the infinite positive x-axis is zero.

To see this, note that the ratio of the lengths (a) from the origin to any finite number to (b) the entire positive x-axis is zero. This is koren's argument that for an even distribution of probabilities the size of the region gives the answer. That part of his argument is not flawed. The flaw is to assign different domains for the choice of A and B. That is, to assume A is bounded while B is not. Since the premise has zero probability, so does the conclusion.

Edited by bonanova

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'Could be over thinking this'

for any 2 positive numbers A and B, there are just 3
possible
relationships

•
• A = B
•
• A > B
•
• A < B

If we assume that the range of numbers is really large then the P [A
=
B]  is very, very low.

If we say that P [A=B] = 0, then there are just the 2 choices [A>B
or
A<B] the relative magnitude of the numbers doesn't matter

and it is simply a matter of which number was chosen first.

A random process is just as likely to pick "small then large" as it is
to
pick "large then small" and the P [A>B] = P[A<B] = 0.5

this holds whether the 2 numbers are (1 and 2) or (1 and a trillion)

Could be over thinking this!

for any 2 positive numbers A and B, there are just 3 possible
relationships

•
• A = B
•
• A > B
•
• A < B

If we assume that the range of numbers is really large then the P [A =
B]  is very, very low.

If we say that P [A=B] = 0, then there are just the 2 choices [A>B or
A<B] the relative magnitude of the numbers doesn't matter

and it is simply a matter of which number was chosen first.

A random process is just as likely to pick "small then large" as it is to
pick "large then small" and the P [A>B] = P[A<B] = 0.5

this holds whether the 2 numbers are (1 and 2) or (1 and a trillion)

Edited by dgreening

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For me, it's not obvious.

Which part?

If you don't understand the first statement, then just look at the definition of probability.

Let me quote from Wikipedia:

"Probability is the measure of the likeliness that an event will occur."

The key word here is measure. Measure is a non-negative value you assign

to each measurable set of your space (the assignment should have some special properties).

No measure - no probability, as simple as that, just by the mere definition.

So let me state it once again: 1) we need to agree upon which subsets of our space are measurable,

2) we need do assign them measure, and only then 3) we can ask about probabilities of different events.

Before steps 1) and 2) probability is not even defined, so questions like 3) are meaningless.

PS Your example problem with rain and plots could be easily resolved by choosing a finite number of sets

that should me measurable and assigning them proper measures. Same procedure cannot be done to solve the original problem.

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Given that both random points lie at infinity and cannot be compared,

the argument of symmetry that holds on any finite portion of the real

axis can be taken to hold in the infinite case.

P[b<A] = 0.5

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Let's start by considering the finite case for a uniform distribution.  Let's consider A and B on an interval [0,S].

Probability of picking point A at the value x is 1/S.

Probability that point B has a value greater than x is (S-x)/S

To calculate the probability for all x, we integrate P(A is at x)*P(B has a value greater than x) over the interval [0,S].

which is INT[(S-x)/S^2] = (Sx-x^2/2)/S^2 from 0 to S.  Plugging in the boundaries, we get (S^2-S^2/2)/S^2

For finite S, we can simply divide out S^2 from top and bottom and we see it simplifies easily to 1/2.

To take the limit as S approaches infinity, we can use L'Hopitals rule twice and we also get 1/2.

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Let's start by considering the finite case for a uniform distribution.  Let's consider A and B on an interval [0,S].

Probability of picking point A at the value x is 1/S.

Probability that point B has a value greater than x is (S-x)/S

To calculate the probability for all x, we integrate P(A is at x)*P(B has a value greater than x) over the interval [0,S].

which is INT[(S-x)/S^2] = (Sx-x^2/2)/S^2 from 0 to S.  Plugging in the boundaries, we get (S^2-S^2/2)/S^2

For finite S, we can simply divide out S^2 from top and bottom and we see it simplifies easily to 1/2.

To take the limit as S approaches infinity, we can use L'Hopitals rule twice and we also get 1/2.

While the final answer doesn't change..

Probability of picking point A at x is 0. I think you meant to say, probability of picking point A less than x. And that would be x/S (not 1/S). Note that A and B can be any points on x-axis, not necessarily at integral distance from origin.

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