Thread: Probability picking a number in an infinite set?

1. Probability picking a number in an infinite set?

what's the probability of picking a number in an infinite set.
EG. out of all the real numbers between 0 to 1 (or an two value for that matter - 0 and 1 are used for simplicity)?

As there is an infinite amount of such real numbers then the probability is 1/infinity. Is this equal to 0?
If so this says that it is impossible to pick a number out of an infinite amount of numbers. Does this imply that for the analogy of snapping a 1m rule (and any other real world analogy), there is a finite number of point the rule can be snapped at?
OR
Does 1/infinity not equal 0? in which case what the devil is it's value? infinetely small but not 0?

Thanks
Jonny

2. Originally Posted by jonny87
what's the probability of picking a number in an infinite set.
EG. out of all the real numbers between 0 to 1 (or an two value for that matter - 0 and 1 are used for simplicity)?

As there is an infinite amount of such real numbers then the probability is 1/infinity. Is this equal to 0?
If so this says that it is impossible to pick a number out of an infinite amount of numbers. Does this imply that for the analogy of snapping a 1m rule (and any other real world analogy), there is a finite number of point the rule can be snapped at?
OR
Does 1/infinity not equal 0? in which case what the devil is it's value? infinetely small but not 0?
Because $\displaystyle \infty$ is not a number $\displaystyle \frac{1}{\infty}$ is not defined and therefore equals no number.
It is simply not defined.

Now it is true that if we randomly pick any number in $\displaystyle [0,1]$ the probably of that number being $\displaystyle \frac{1}{2}$ is zero.
That does not mean that $\displaystyle \frac{1}{2}$ cannot be picked. It can with probability zero.
This is a proper question.
Suppose we randomly pick any number in $\displaystyle [0,1]$, what is the probability that the number is within 0.2 units of $\displaystyle \frac{1}{2}$?
Well the answer is $\displaystyle 0.4$ because we asking $\displaystyle P(0.3\le X \le 0.7)$.

Does that make sense to you?

3. Thanks for the reply..yes I understand what you are saying.

Wouldn't the probability of picking an real number in [0,1] be 1/infinity? meaning 1/2 would have this probability also. so are you saying that 1/infinity=0?

And isnt "a probability of 0" another way of saying that the event is impossible?

Thanks
Jonny

4. There is a difference in infinities.
Countable sets can have positive measure, for example the geometric rvs. Discrete rvs are those that have finite or countable infinite support.
The uncountable sets, such as (0,1) are too big, so we need to define a density function. We use f(x) as thaat function, it equates area to the probability of that event.

5. Originally Posted by jonny87
Wouldn't the probability of picking an real number in [0,1] be 1/infinity?
$\displaystyle \infty$ is not a number
There is no such an expression as $\displaystyle \frac{1}{\infty}$ it is total nonsense.

Originally Posted by jonny87
And isnt "a probability of 0" another way of saying that the event is impossible?
Absolutely not.
If we randomly pick any number in $\displaystyle [0,1]$ the probably of that number being $\displaystyle \frac{1}{2}$ is zero.
But it should be clear to you that it is possible that $\displaystyle \frac{1}{2}$ can indeed be picked.

6. Thanks,

I don't understand what you mean by terms like rv and "infinite support". Could you explain please?

Are you saying that it's impossible to calculate the probability of an event in an infinite set? But that this doesn't mean the event is impossible?

Thanks
jonny

7. Ah ok..this confused me as I have always thought that probability of 0 means impossible and that a probability of 1 means certain. So are you saying that this isn't true?

Is it that the probability tends to 0 rather than it actually equals 0?

Thakns
jonny

8. Originally Posted by jonny87
Is it that the probability tends to 0 rather than it actually equals 0?
The reason being that the level of that question suggests that you should never misunderstand the concept of infinity. Nevertheless you have misused the concept repeatedly.

9. Originally Posted by jonny87
Ah ok..this confused me as I have always thought that probability of 0 means impossible and that a probability of 1 means certain. So are you saying that this isn't true?
Just to underline what Plato has been saying, probability 0 does not mean impossible, and probability 1 does not mean certain.

For example, if a real number between 0 and 1 is chosen at random, there is a probability 1 that it will be irrational, and a probability 0 that it will be rational. But to make rigorous mathematical sense of that statement, you need to know some measure theory.

10. Originally Posted by Opalg
Just to underline what Plato has been saying, probability 0 does not mean impossible, and probability 1 does not mean certain.

For example, if a real number between 0 and 1 is chosen at random, there is a probability 1 that it will be irrational, and a probability 0 that it will be rational. But to make rigorous mathematical sense of that statement, you need to know some measure theory.
There is also a problem with: "real number between 0 and 1 is chosen at random" (I will assume some form of uniform distribution is meant).

It is difficult to see what this could mean if you have a basic not very mathematical understanding of probability.

With a sophisticated understanding of (mathematical) probability this does make some sense, but it is difficult to see how it would arrise in practice other than as an idealisation of a real experiment.

When I see an expression like "a real number between 0 and 1 is chosen at random" I always ask myself "how would I do that?" and "how would I know which number has been choosen?"

CB

11. Im have just finished my first year at univeristy.

I realised that infinity is not a number and so cannot be used as one, but I didn't know a better way of sayin what I meant.

THanks
Jonny

12. Thanks for the replies

Opalg:

CaptainBlack:
Wouldn't the distribution of real number in [0,1] be uniform anyway?
This was meant as a purely theoretical question as I see that in practice there would be problems.

Thanks
jonny

13. Originally Posted by jonny87
CaptainBlack:
Wouldn't the distribution of real number in [0,1] be uniform anyway?
This was meant as a purely theoretical question as I see that in practice there would be problems.
From the context I think we have all assumed you meant the uniform distribution on [0,1], but it is necessary that the distribution be specified.

CB