Question about infinite probability?

NeptuniumBOMB

i was just wondering (theoretically thinking), if I had to pick "option 1" from an infinite amount of options and my pick was random and I have an infinite amount of time (kind of like taking a ball from a bag, don't know which one is which) but I have to get exactly "option 1" and not any other.
Is their any physical chance that i will ever get the option i want or is it impossible because an infinite amount of options means that your probability is to small to exist?

Just wondering this, it isn't part of my homework or textbook.

Millennial

In general, if there are infinitely many events and you are considering the probability to select finitely many, then the probability is zero. Else way, it is the density of the set of selections in the set of events.

Given that you have infinite time though, you will absolutely pick option 1. Same goes with all other finite sets of options. If you had finite time, then we could not say the same.

micromass

Everything depends on the actual probability distribution. How likely is it to pick option 1??

Also, are you working with countable infinite or uncountable??

These things are needed to give a suitable answer to your question.

haruspex

The trouble is you cannot have a uniform distribution over a countably infinite set. All probabilities would be zero, and the total probability would be zero, which is not valid.
You could redefine it as a limit problem as the size of the set tends to infinity. To complete the definition, you need to specify how the time (number of trials) available depends on the set size, n, and whether the ball is replaced after each trial. E.g. if you specify n trials, with replacement, the probability of success tends to 1 - 1/e.