Suppose the random variable

has the standard normal distribution,

. Pick any real number, say

. What is the probability of the event

? The answer is

, i.e.

, as any high school student will tell you. So the event

has probability

, that is

*almost surely*.

Do you think this means that it is impossible for

to be exactly

? You may say yes, and you may be right, but then the same is true for any other value. You are left wondering how it is possible for

to take

*any* value whatsoever, given that it's impossible for

to take any particular value.

Do you begin to see the difference between the terms

*surely* and

*almost surely*?