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