Understanding Probability: Machine Choosing Digits & Blinking Intervals

[aerosmith]

i recently thought of something regarding probability. and i have no clear answer to the question, i stumbled onto this site and thus needs all your help.

if there is a machine that chooses a digit between 0 to 9 , what's the probability it chooses 1? well, my answer is 0.1, fact is, 0.1 is the probability the machine will choose 2,3,4,5,6,7,8,9,0.

here comes the problem, if the machine is asked to choose between 0 to infinity, the probability for a number to be chosen is now 1/inf=0 , then i wonder if it makes any sense or not.

so please enlighten my stupid brain, i know this is simple for u all, but thanks anyway.




another question along the same lines is imagine a machine that blinks every 1-9 seconds, thus on average, the machine blinks every 5 seconds. what is the machine is made to blink every 1 to inf seconds, the machine blinks every ___ seconds?
i am really confused about this one. i thought about these two questions when i was learning probability.

once again, thanks for all u ppl help

 

[John Creighto]

if there is a machine that chooses a digit between 0 to 9 , what's the probability it chooses 1? well, my answer is 0.1, fact is, 0.1 is the probability the machine will choose 2,3,4,5,6,7,8,9,0.

This would be correct if the probability of choosing any given number is equal.

here comes the problem, if the machine is asked to choose between 0 to infinity, the probability for a number to be chosen is now 1/inf=0 , then i wonder if it makes any sense or not.

If you could randomly select a number between zero and one the probability of selecting any number would be zero. Yet you must select something. Just because something is unlikely doesn't mean it is impossible.

another question along the same lines is imagine a machine that blinks every 1-9 seconds, thus on average, the machine blinks every 5 seconds. what is the machine is made to blink every 1 to inf seconds, the machine blinks every ___ seconds?
i am really confused about this one. i thought about these two questions when i was learning probability.

once again, thanks for all u ppl help

Well, if all time delays between 1 to are infinity are equally probable then you could be waiting a very long time for the machine to blink.

 

[EnumaElish]

if the machine is asked to choose between 0 to infinity, the probability for a number to be chosen is now 1/inf=0

You seem to have started your question with a discrete uniform distribution defined over integers from 0 to 9. If you extend its range to the entire set of nonnegative integers then the probability for any number being selected will be zero. The expected value (the mean) is then defined as \sum_{i=0}^{+\infty} xp(x) = \sum_{i=0}^{+\infty} 0 = 0, but it is intuitively clear that the expected value cannot be zero. (In fact, the "intuitive" value of the mean is +\infty/2 =+\infty.) So you are right, it does not make sense to define discrete uniform probabilities over a (countably) infinite set.

 

[EnumaElish]

In my previous post, \sum_{i=0}^{+\infty} xp(x) should have been "undefined" and not 0.

\sum_{i=0}^{+\infty} xp(x) = \sum_{i=0}^{+\infty} (x \times 0) = 0 \times \sum_{i=0}^{+\infty} x = 0 \times (+\infty) = \text{undefined.}