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Kind of an interesting one....

Kind of an interesting one....

Posers and Puzzles

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What is the probability that a randomly selected rational number (fraction in lowest terms) has an even denominator?

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Originally posted by royalchicken
What is the probability that a randomly selected rational number (fraction in lowest terms) has an even denominator?
Again, I am lost in terminology.....I apologize. What do you mean by rational number?

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No problem. I'll try and be clear. A ratioinal number is a fraction in lowest terms with whole numbers as the numerator and denominator. For example, 3/2 is a rational number.

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Originally posted by royalchicken
No problem. I'll try and be clear. A ratioinal number is a fraction in lowest terms with whole numbers as the numerator and denominator. For example, 3/2 is a rational number.
I see. Does this mean that for you puzzle all numbers are written as fractions. eg. 1 = 1/1

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Originally posted by Cheshire Cat
I see. Does this mean that for you puzzle all numbers are written as fractions. eg. 1 = 1/1
Exactly. 1=1/1. That is a crucial bit too...you're almost halfway there 😀

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Originally posted by royalchicken
Exactly. 1=1/1. That is a crucial bit too...you're almost halfway there 😀
Does this problem have a cap, or a number that it can't go above?

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No...of all the rational numbers, what is the probability that a randomly selected one has an even denominator?

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Originally posted by royalchicken
No...of all the rational numbers, what is the probability that a randomly selected one has an even denominator?
Ok...here goes...The number you have to choose from are 1/1 to infinity over 1 or infinity. Due to another thread I know that all sets of infinity are equal so the choices 1/2, 3/2, 5/2.... are infinite So the chances theoretically are infinity over infinity or one. Did I get it or am I missing something?

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I'm afraid not sir. Think of all of the rationals less than, say, 10, with denominators less than, say, 10. Count how many have even denominators. Then make your parameters bigger and see what patterns emerge.

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Originally posted by royalchicken
I'm afraid not sir. Think of all of the rationals less than, say, 10, with denominators less than, say, 10. Count how many have even denominators. Then make your parameters bigger and see what patterns emerge.
I'm afraid I am still missing something....if you count even denominators...1/2, 1/4, 1/6, etc....you still come up with an infinite set of numbers, right. So therefore my answer would stand. What am I missing?

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I'm trying not to be too technical here. Thing is, you're right: there is an infinity with even denominators, and also an infinity with odd denominators. My question could be phrased "For each rational with an even denominaotr, how many rationals with odd denominators are there?"

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Originally posted by royalchicken
I'm trying not to be too technical here. Thing is, you're right: there is an infinity with even denominators, and also an infinity with odd denominators. My question could be phrased "For each rational with an even denominaotr, how many rationals with odd denominators are there?"
Alas, that gets back into the infinity question....it would still be a 1 to 1 ratio due to the properties of infinity. Any other puzzles?

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Interestingly it would not; tis not a 1/1 ratio. But I'll leave it to someone to solve. I'll think of another.

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Originally posted by royalchicken
What is the probability that a randomly selected rational number (fraction in lowest terms) has an even denominator?
This can't be solved.
The problem is the infinity. The same problem you get is when you look at the chance of picking an even number out of al numbers. The problem then is, that you can't say that half of the numbers are even. cause if you order them differenty like: 1, 3, 2, 5, 7, 4, 9, 11, 6, 13, 15, 8, 17, 19, 10, ... then you still got all of them and you have a 2 to 1 ratio. If youonly look at the numbers less then 10^1000 then you can say half of them are even.

What you can say is that the probability of picking a rational out of the reals equals 0.

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Fiathahel, it can indeed be solved, although I see where you are coming from. I sent the answer to Cheshire Cat with the smallest germ of an argument.

It is an interesting one to mess with, and I'll see if anyone posts the answer in the next week or so. If not then I'll post it.

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