Rev. Green | 11:12 Sat 06th Jun 2020 | Science
An infinite number of points A are placed at random on an infinitely long line. A second infinity of points B are also placed at random on the line. How many B's coincide with one or more A's on average?

My gut feeling is that the answer should be zero but I am sure there's a better way to answer this than relying on points/line = countable/uncountable -> 0
11:51 Sat 06th Jun 2020
Son:

I think you could apply the infinite pairs argument, so infinite
I'm still not sure we are interpreting the question in the same way.
I took it we are placing an infinite number of points which can be labelled Ai for i=1 to infinity, i.e A1, A2, A3,....A999999, …
and then labelling another infinite number of points Bi for i=1 to infinity. i.e. B1, B2, B3, …..B999999, …

Now when we count the number of pairings are we only looking for A1=B1, A2=B2, A3=B3,....
or are we looking for A1= B1 or B2 or B3 ….B9999999....then A2= B1 or B2 or B3 ….B9999999.... then A3 = B1 or B2 or B3 ….B9999999.... etc etc
If there are points A1 and A2 on the infinite line, I read it that there could be an infinite number of points B1,B2, B3 etc before, on or after points A1 and A2.
Question Author
As there are aleph one positions on the line, and only aleph null points, I think the answer is zero. Perhaps a better question is "What is the average distance between two consecutive points when an infinite number of points are placed at random on an infinite line?"
My son:
How do we know there are one positions and null points?
Question Author
Sorry, I should have said aleph-one positions (i.e. An uncountable number of positions) and aleph-null points (i.e. A countable infinity of points). The aleph numbers represent differing degrees of infinity. The number of integers is aleph-null, and there are the same number of even numbers, the same number of perfect squares, primes, rational numbers, etc. But there are substantially more real numbers, so they are represented by aleph-one, which is the next higher infinity.
The probability of a point B being randomly placed at the same point on a line as that of point A is zero; since this is being applied an infinite number of times – the total number that will coincide is zero times infinity.
I can see that, hymie, if A1 has to match B1 and A2 has to match B2.
But if there's an infinite number of point Bs then I feel the probability of a single point A coinciding with ANY one of the Bs won't be zero.

The infinite line must be made up of an infinite number of points. Would that number of points not equal the number of Point As and the number of Point Bs?
No, sadly. As Rev. Green's hinted there are different sizes of infinity. I'll try to clarify this later if you're interested, but in short it's the difference between trying to count "1, 2, 3, 4, ... " and "0.0000000000000000000000...". In the first place you never stop counting but at least you can kind of see that you can make progress, in the second you never even get away from 0.
Well according to deep thought the answer to the ultimate question and the whole of the universe is indeed 42 but of course coldplay have a different version
The answer is not 42. That much I know.
Well tell that to deep thought then tomas
And I thought R&S was one for conjecture???
Most theorems in mathematics started life as conjectures, though.
Maths was never my strong point...
It can take many years at university to be able to talk confident conjecture.
I can also see zero as the answer, although I am still not sure we are interpreting the question the same way. It's zero I think if Ai has to match Bi
If the first point A1 is placed at say 3.5 exactly there is an infinite number of other points that B1 could be placed at. Even getting close at 3.50000000000000001 say isn't good enough.

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