1. From the Big Bang onward there is infinite time.
2. In the lottery too, sometimes the win comes early….

1. ??? That’s a refreshing innovation in mathematics (not related to physics): there is an event that happened at some point, and after it an infinite amount of time has passed.
2. So according to your approach, any statistically rare event says nothing at all. For example, if in rolling a die you get 6 a hundred times in a row, you should not conclude that the die is unfair, because sometimes such a sequence happens by chance right at the beginning. And so on. Do you really think that is reasonable? The fact that something is possible with some probability is not a relevant consideration. One has to think about it in terms of hypothesis testing: you ask yourself which of the two hypotheses is more probable—that it happened by chance right at the start, or that there is a guiding hand. Both possibilities can of course occur, except that the first has a negligible probability. Therefore the second is preferable.

1. What I meant to say is that there could be infinite time after the Big Bang, not that right now we are standing at “infinity.”

2. Assuming the die were rolled infinitely many times,
I assume the Rabbi agrees that in the case the Rabbi described, there would be nothing especially surprising.

So why should it matter whether it was created at the “beginning” of the infinite timeline or at the “end” of the infinite timeline.

Just like people win the lottery every day, etc.
The Rabbi is invited to watch some atheist videos—they demonstrate this very well. With all kinds of effects and so on.

Dear Rabbi, I didn’t understand your rejection of infinite time before and after the Big Bang.
And after the Big Bang until the formation of life because of the protein chain—how do you know how long it took?
I also didn’t understand how one can speak about “time” at all, long before the luminaries were created (however they were created is not important right now).

1. There is no infinite time, neither before nor after. Infinity is a potential concept. But even if we are talking about a long time (and not an infinite one), one can speak of a small probability being realized. However, we are not familiar at all with any process of universes and life coming into being out of nothing, so the duration of time does not provide a solution. Especially since the laws within which all this takes place themselves require explanation. All this appears in my article here on the site, and perhaps it would be worthwhile for you to read it:

מבט שיטתי על יחסי אבולוציה ואמונה

As for a small probability being realized at the beginning of time, I already gave you an example from die results. Think about it.

I know the atheists’ arguments very well, and their demonstrations are usually wrong and misleading. In general, probability and statistics are very confusing fields for laymen, and it is very easy to mislead and to make mistakes.

Moshe, time is not connected to the luminaries. Cosmic cycles only measure time, but they are not the essence of time. You too are invited to read my above article.

I didn’t understand what the difference is between the sequence coming out at the beginning or at the end, in the die parable.

Also, life did not exactly arise at the beginning, but rather fairly soon after the beginning.

Also, what the Rabbi said in the link, about the fact that we don’t know of other universes, etc.—I’m talking about this universe having infinite time.
And I didn’t understand the difference regarding potential infinity, etc.?

Is the Rabbi saying that the atheists’ illustration in the videos with the lottery is incorrect?

The lottery example is complete nonsense. Just plain misunderstanding. In general, I know their examples and arguments, and most of them are simply beside the point. I referred you to the article and the book; look there.