The probability of an event occurring at an exact moment in a continuous time framework is defined as zero, as demonstrated in the discussion regarding a robot pressing a button at a random time within a two-minute interval. The reasoning is based on the concept that with infinite possible events, the probability of hitting a specific point, such as exactly one minute, is 1/infinity, which equals zero. However, this does not imply that the event cannot occur; once the button is pressed, the probability of it happening at that exact moment becomes one. This distinction between theoretical probability and actual occurrence is crucial in understanding continuous random variables.
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quadraphonics said:Indeed, the cost of including infinity in a number system is that certain basic expressions must remain undefined, when they include infinity. But so what? They're still defined as usual for every finite number, and the expressions where infinity is sensible (like 1/\infty=0) are still well defined.
There is also a finite number that does not work in many basic algebraic manipulations, but is nevertheless included in many standard number systems.
Archosaur said:Can you give me an example of a finite number that I can't multiply both sides of an equation by?
Archosaur said:I would agree that the limit as n -> infinity of 1/n = zero
But I'm having a hard time with this 1/infinity = 0 thing...
quadraphonics said:\infty, as it is used in the extended reals, is essentially a shorthard for that type of limit expression.