An infinite debate?
March 23, 2011 10:54 PM Subscribe
Got into a debate with a friend regarding a math question - I think our debate reflects his misapplication of finite math reasoning to infinities, but I'm having trouble dredging up what I learned in set theory 10 years ago. Imagine you have a bag that has all of the real numbers from 0 to 2, let's say. What are the odds that if you reach into the bag you pull out a number between 0 and 1.5 ? 75%? 50%? Why? Depending on the answer to this I may have followup questions, but I'll leave it here for now.
posted by slide to Education (9 answers total) 5 users marked this as a favorite
Screw it - followup question now. Same question for 0 to 180, with 0 to 90 being what you are trying to pull?
Now imagine the bag contains all triangles with two sides of 1 unit length and an angle between those two sides of 0 to 180. Thus, the bag contains an infinite number of triangles representing all possible acute, right, and obtuse triangles. What are the odds that if you reach into the bag you pull out an acute triangle? If you consider the angle between the two sides as the determining factor, it is between 0 and 180 and anything less than 90 is acute. But if you consider the length of the far side to be the determining factor, it is between 0 and 2, and anything less than sqrt(2) or ~1.4 is acute. So for the former, it is equivalent to the 0 to 180 question; for the latter, the 0 to 2 question. Yet the bag of triangles is the same, so the probability has to be the same no matter how you look at the problem, right? Or am I missing something?
My contention is that if you have two disjoint infinite sets of the same cardinality, then if you combine those sets, the probability of selecting an element of either set at random will always be the same. That seems odd in the context of the 0 to 2 problem above, so I thought I'd appeal to a higher cardinality of intellects.