Please answer the following math/philosophy question

classic undefined problem.

You are asking what is infinity/infinity = undefined

Sudden, unexpected & horrific flashback to Philosophy 2010 at LSU.

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1. A time line that is infinite

2. Events that are unlikely to happen often but have a non zero probability of occurrence

Is it true that all events with a non zero probability will eventually occur given an infinite amount of time?

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This is easy.

Infinite time and something that is extremely, extremely rare, but not of zero probability.

Well, since you yourself have stated that this has a non zero probability (meaning it has a chance of happening), and you have infinite time, than it will eventually happen.

There isn't a question of will it happen, the question is when will it happen.

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Is this like the monkey hitting random keys being able to type Shakespeare given long enough time to type?

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Yes, and given the script of a play by Shakespeare in hand, which is more likely: that it was designed intelligently or created at random by a monkey?

yes

suppose Event X has probability Y of occurring in one attempt.

probability of X = Y
probability of not-X = 1-Y

in two attempts:

probability of X twice = Y^2
probability of not-X = (1-Y)^2
probability of X once = 1 - Y^2 - (1-Y)^2
probability of X *at least once* = 1 - (1-Y)^2

in Z attempts:

probability of X Z times = Y^Z
probability of not-X = (1-Y)^Z
probability of X *at least once* = 1 - (1-Y)^Z

this last one is the one we need, because we can cleanly separate all possibilities into two groups: X happening at least once, and X never happening

so, we have 1 - (1-Y)^Z
per condition (1), Z goes to infinity
0 < 1-Y < 1, so as Z goes to infinity, (1-Y)^Z goes to 0

1 - (1-Y)^Z = 0.999...
0.999... = 1
1 - (1-Y)^Z = 1
probability of X *at least once* = 1 - (1-Y)^Z = 1

therefore we conclude that yes, given an infinite timeline, any event with a non-zero probability will eventually occur

EDIT: extending this a bit to cover one clause that might make some people confused

0.999... = 1

the classic proof of this is with 1/3

1/3 = 0.333...
1/3*3 = 1
0.333...*3 = 0.999...
0.999... = 1

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This is easy.

Infinite time and something that is extremely, extremely rare, but not of zero probability.

Well, since you yourself have stated that this has a non zero probability (meaning it has a chance of happening), and you have infinite time, than it will eventually happen.

There isn't a question of will it happen, the question is when will it happen.

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Not really. There's only the possibility that it could happen. Doesn't mean that it will. Obviously not everything that can possibly happen eventually happens. Some things never happen.

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therefore we conclude that yes, given an infinite timeline, any event with a non-zero probability will eventually occur

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So my aunt that never had kids before she died will someday actually have kids even though she's already dead?

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So my aunt that never had kids before she died will someday actually have kids even though she's already dead?

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a dead person has probability zero of having kids

your aunt, when she was alive, did not have infinite time for the event of her having kids to occur

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does this have to do with the megathread about evolution?

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Yes

I always here that it's "severely unlikely" that X could happen.

Well I contend that when you're dealing with a few billion years a lot of things that are very abnormal and rare become more and more likely to occur.