While laying in bed the other night I came up with two interesting math questions.
1) Choose
integers at random. Multiply them all together. What is the probability that the product of all the integers you chose is odd?
2) Choose
I like these problems a lot because they have an intuitive answer that is not right, they help show the importance of parity arguments, and the second one brings up a really interesting mathematical concept about a probability being infinitely small.
Just some food for thought.
What is your distribution for “random integer” or “random prime number”? With infinite sets it can’t be uniform, and there isn’t another default choice, so you have to specify.
It’s true, but I came up with these problems more as thought exercises than things to solve. I suppose I would have to specify if I want people to arrive at a specific (right) answer, but I’d these questions to get people to think about the ideas (especially regarding infinite sets) rather than solve them.
Although I think that for the first question you can arrive at a solution without me specifying.
Well, if odds and evens are equally likely, then obviously 2^{-n} for the first question. If odds have probability p, then p^n.
For the second question, it all depends on the probability of choosing 2 as 1 (or more) of the primes. If the random choice is i.i.d. (independent, identically distributed) and the probability of choosing 2 on any one draw is p, then (1-p)^n.
As probability questions and infinite set questions go, these don’t lead to any very deep questions or insights. I think that there are much more fun combinatorics questions to lose sleep over.
Hmm… I suppose I was going at the problems with a different (less-mathematical?) approach. I don’t exactly know the conventions of probabilities with infinite sets, so I made it up in my head. I figured that choosing random numbers yields a 50% chanceof being even and 50% chance of being odd, so the probability of the product of the numbers being odd would be
.
For the second question, I figured that there is one even prime and infinite odd primes, so the probability of ever choosing an even prime number would be 0% making the probability of the product being odd 100%.
I think that the issue here is that you have significant knowledge in this field of mathematics and don’t see any deep questions and insights (as they have already been answered for you). I on the other hand am stumbling around in the dark trying to figure things out. Perhaps the problems are in my ZPD but far below yours.
Also, solving these problems are not what I’ve been losing sleep over. Rather, formulating the question has been quite thought provoking.
The problems of dealing with infinite sets are interesting. Your approach (probability of choosing 2 is 0) has an obvious problem: by the same reasoning the resulting number will have zero probability of being divisible by any particular prime.
Choosing at random from an infinite discrete set can’t be done uniformly. The sum of the probabilities of the individual choices has to be 1, so they can’t all be 0.
For the primes, P(2)+P(3)+P(5)+P(7)+… =1. Obviously, for this sum to converge, the probabilities can’t all be the same, but they can all be non-zero, which satisfies at least one intuitive expectation of “choosing at random”. You correctly observed that what matters is P(2), but then made the mistake of assuming that P(2) had to be zero.