infinitely small number x infinity?

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derekw

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OK, so I don't even know if I have a correct understanding of "infinity", but here is my question:

If something has an infinitely small probability and you times that probability by infinity, what would you get?
so: infinitely small x infinity =?

Are infinitely small probabilities even possible?
Does "multiply by infinity" even mean anything?

Thanks for any answers!
 
You really can't use infinity in regular arithmetic that way. At least, not in the real numbers. In non-standard analysis, there are alternate number systems, such as the hyperreal numbers, where both infinite and infinitesimal values exist. However, even then you have to be careful with how you operate with them.

For the purposes of probability theory, an infinitesimal probability is the same thing a zero probability. This is why we consider the probability on intervals rather than countable sets when dealing with continuous random variables.
 
"infinity" x "zero" ... as a limit is an indeterminate limit . This means that there are infinitely many possibilities for the answer.


Example:

lim (1/(x+1)) * (5x+2) as x→∞
the first factor goes to zero
while the second factor goes to infinity

lim (5x+2)/(x+1)
= lim 5/1 = 5 using L'Hopital's rule.


now
lim (1/(2x^2 + 1)) (6x^2 + 5)
= lim (6x^2 + 5)/ (2x^2 + 1)
= lim (12x)/(4x)
= 3

thus depending on the nature of both the numerator and the denominator ...
or in other words, on the nature of the expressions going to zero and infinity, then the answer can be anything. (even zero or infinity).
 
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