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:) More Math. Worse than .9999...=1

DrPizza

DrPizza

Administrator Elite Member Goat Whisperer
If a number between 0 and 10 is chosen completely randomly, with all numbers having an equal probability of being selected, well, then, the probability of selecting a rational number is Zero!
+
 
The probability of choosing any given number approaches zero, if you are choosing from a continuous range between any X and Y. There an infinite number of both rational and irrational numbers in any range.
 
That's basic probability regarding continuous variables.

Just like the area of each inifitely small rectangle when you integrate is 0, but the sum of the 0s is the evaluated integral (and the total area).
 
Originally posted by: dighn
Originally posted by: ActuaryTm
Given event A being this thread proving successful and popular, it can also be said:

P(A) = 0
Click to expand...

😀
Click to expand...

🙂

Hey, since so many people in the .999... thread couldn't comprehend infinity, I figured I'd toss in that there are different sizes of infinity.
 
I don't think people are getting it.

There are an infinite number of rationals between 0 and 10 people. DrPizza is saying that your chances of picking any of those infinite number of points is zero. He's not saying to pick one (say 3/4) and asking what your chances of picking that number are. It's actually quite a big statement. 🙂
 
Originally posted by: silverpig
I don't think people are getting it.

There are an infinite number of rationals between 0 and 10 people. DrPizza is saying that your chances of picking any of those infinite number of points is zero. He's not saying to pick one (say 3/4) and asking what your chances of picking that number are. It's actually quite a big statement. 🙂
Click to expand...

:Q

I'm just making sure I get this when I say what's below:

For all rational numbers between 0 and 1:

Say you take 1/x as your first rational number, where x = 1. Then you do this for x =2, 3, 4, "going on forever." And that's just a small set of all numbers between 1 and 0. So you can basically create an infinite amount of rational numbers just between 0 and 1.

Then you do the same concept for every number between 1 and 2, and so on.



Since there's an infinite number of rationals, the probability of picking any single rational number out of what is effectively an "infinite" amount of numbers, the probability is 0.

Is that a correct way of thinking about it?
 
Originally posted by: BigJ
Originally posted by: silverpig
I don't think people are getting it.

There are an infinite number of rationals between 0 and 10 people. DrPizza is saying that your chances of picking any of those infinite number of points is zero. He's not saying to pick one (say 3/4) and asking what your chances of picking that number are. It's actually quite a big statement. 🙂
Click to expand...

:Q

I'm just making sure I get this when I say what's below:

For all rational numbers between 0 and 1:

Say you take 1/x as your first rational number, where x = 1. Then you do this for x =2, 3, 4, "going on forever." And that's just a small set of all numbers between 1 and 0. So you can basically create an infinite amount of rational numbers just between 0 and 1.

Then you do the same concept for every number between 1 and 2, and so on.



Since there's an infinite number of rationals, the probability of picking any single rational number out of what is effectively an "infinite" amount of numbers, the probability is 0.

Is that a correct way of thinking about it?
Click to expand...


It is correct that the probability of picking any single (or prespecified) rational number is 0, but as silverpig said, the statement says more than that, i.e. the probability of picking a rational number (doesn't matter which one) is also 0.
 
So basically this is a .9999... = 1 thread, but you reversed and disguised it so people wouldn't recognize it.
:thumbsup:
 
Originally posted by: DrPizza
Originally posted by: dighn
Originally posted by: ActuaryTm
Given event A being this thread proving successful and popular, it can also be said:

P(A) = 0
Click to expand...

😀
Click to expand...

🙂

Hey, since so many people in the .999... thread couldn't comprehend infinity, I figured I'd toss in that there are different sizes of infinity.
Click to expand...



Or to make this a little more personal, where A is the thread proving successful and popular, and B being a thread started by DrPizza

P(A|B) = 0

😀

j/k dr!
 
Hold, on, wouldn't the probablity of that just be the total number of numbers in the set(infinity) divided by the number rational numbers in the set (infinity), which means that the answer isn't 0, the answer is meaningless.
 
Originally posted by: Kibbo
Hold, on, wouldn't the probablity of that just be the total number of numbers in the set(infinity) divided by the number rational numbers in the set (infinity), which means that the answer isn't 0, the answer is meaningless.
Click to expand...

Well, the answer is an indeterminate. This would be a limit, use L'Hospital's to figure it out. Just depends on which one goes to infinity quicker, if I remember correctly.
 
Originally posted by: DrPizza
If a number between 0 and 10 is chosen completely randomly, with all numbers having an equal probability of being selected, well, then, the probability of selecting a rational number is Zero!
+
Click to expand...

Makes sense...
 
Clearing it up more by adding n00bness:

There are rational numbers, and there are irrational numbers. If there were an equal number of rationals to irrationals, your chances of picking a rational number in that range would be 50%. The random number you pick could be either rational or irrational right?

Now, there are more irrational numbers than there are rational numbers (infinitely more... and it gets weird...). So now, when you pick a random number in a range you still have two possible outcomes: You can pick a rational number, or you can pick an irrational number.

Let's assume that there is a single basket with apples and oranges. There are twice as many apples as oranges. Your chances therefore of picking an apple are twice as great as picking an orange, so the associated probabilities are 2/3 and 1/3...

There are an infinite number of both irrationals and rationals, but there are infinitely more irrationals than rationals. Since no number is preferred over any other, your chances of picking an irrational must be infinitely times as great as picking a rational number, and hence the associated probabilities are 1 and 0 🙂
 
Originally posted by: JohnCU
how?

because there is an infinite amount of irrational numbers between 0 and 10?
Click to expand...

not quite.

because the irrationals are a larger infinity than the rationals; specifically, the irrationals are uncountable.
 
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