Did Isaac get it wrong in the iron stars episode?

In that episode, Isaac claimed, "Over an infinite period of time, anything that can happen will happen...". I agree that if the probability of that thing that "can happen" is static, then it will happen. What if the probability of that thing is not static but decreases over time? Will that thing happen eventually?

Imagine this game. You have a magic die that begins with 6 numbered sides. You win the game if you roll a 1. At the start of the game you have a 1 in 6 chance of winning. If you don't win the game after the first roll, the magic die increases by 1 side so on the next trial you have a 1 in 7 chance of winning. If you go on to a third trial the die has 8 sides, On the forth trial it has 9 sides and so on. Is it possible to roll such a die an infinite number of times and never roll a 1(win the game)? Its always possible to win the game but does an infinite number of tries guarantee you will win it?

Admittedly the above thought experiment is contrived. Nevertheless similar situations do happen in real life. Can a roulette player who starts a quest to quit gambling when he/she is even when they are currently down by $10000 ever have a winning steak that takes them back to even if they play the game for eternity? It could happen, but is it guaranteed? If they play for an eternity they will certainly have winning streaks that clear $10000 in losses. The problem for our degene gambler is while he/she is chasing that streak his/her losses our very likely to grow. After a billion spins our roulette player might have to run into a 100 billion dollar winning streak to get back to even and that is a much harder thing to accomplish..

What if the probability of that thing is not static but decreases over time?
To be absolute, that depends on the specific ways in which the "probability[...] decreases over time". So Will that thing happen eventually? = not necessarily at all.
Its always possible to win the game but does an infinite number of tries guarantee you will win it?
(This site doesn't have the capability to deal with equations so you want the specifics i can write it down and send a pic of it over the Discord.)
In this example, you might be tempted to consider what happens when you are on a 1000000-ish losing streak and think that if you were to continue like that, even though the chances of finally succeeding would be small yet wouldn't be exactly zero. Then use that argument, over a specific case, to justify that you could always win in the end or something. However that's not exactly how "infinities" work. You can totally have sums or products (here the products of the probability of wining/losing at each step) that are infinite and yet converge to a finite number. If you are rusty on indeterminate forms you can skip the computational part and just prove that the probability of not winning on the first try then not winning on the second ... and so on (product of increasing fractions getting closer to one at each step) is bounded by some other probability. Here to simply answer you question of am i guaranteed to win you can show that the infinite product is positive and so you can totally never win. Equivalently, you can try to prove that the possibility of winning your game is less than one.
I really can't answer your next proposal of game properly without hand-waving with just words, but i hope you get the idea.
Now you didn't have to choose such examples, since i presume you were at first thinking about the quantum tunneling thingy that is supposed to get all stars all "rusty" in the end. If we make the egregious approximation of only considering closeness of the two atoms and their "energy levels" (more hand-waving i know sry), then the probability of "protons" (to keep things simple) within a dead star to quantum tunnel and form an iron atom will decrease. Think of the extremes: the first two "atoms" combing will most likely be two close to each other. But now imagine the last step of completely transforming the dead star into a ball of iron: most likely the two last atoms would be pretty far from each other. So you can have couples of atoms that are in kissing distance and other that could be diametrically separated (so miles apart from each other). You can consider that the energy variations would have faded away and been thermally dissipated with the infinite temperature sink/source of the background universe and claim that these variation are negligible but the distance between the two "atoms" is not a constant, and most likely decreases if i may add.
(saying most likely like that is light-years away from good probability explanation but i hope you get my point since in any case the probability of combining this way varies with time and is there not constant and need to be treated a little more cautiously).
Did Isaac get it wrong in the iron stars episode?
Now did we lie to you? not really. it could be that probabilities are in such a way that the amount of dead stars not fully "ironed" end up being very small compared to the fully "ironed" ones. just like in your dice game the chances of winning aren't that bad at all. But then does it really matter if only 78% of the dead stars are fully ironed while the rest of them still have some trillions of not-iron-atom? No it doesn't change the main idea, and that's what matters. In the videos IA isn't a scientist talking to other scientists, he is a scientist talking to a very broad public made of people who may never even have taken a single course in probability, astronomy, QED, or anything "sciency" so yeah we do need to gloss over some details when summarizing:claiming stuff if we don't want to bore our audience to death.
I hope you can agree that killing your audience is never a good PR move :p

I referenced the Iron Stars episode only because that is where I heard Isaac make a claim that strikes me as being wrong and I remembered the specific episode. The claim being: over infinite time lines anything that can happen will happen. To be fair, I hear that claim fairly often so If it is an error, it is a common one. In most circumstances, like the transmutation of the atomic constituents of dead stars, it is probably true that given enough time they will all turn into iron.

I do seemed to remember an episode where Isaac was discussing how in the future we could make copies of ourselves. I recall him making a comment along the lines that if enough time passed eventually some series of calamities would come along which did all your copies in; the ability to make copies of yourself, doesn't bring you some kind of immortality. I wish I remembered what specific episode that was so I could review it again for the sake of this discussion. The ability to make copies of yourself and spread yourself throughout the cosmos means it is plausible that over time the probability of some random set of calamities doing all of your copies in decreases over time. If the probability decreases over time I don't think it is a given you'd be wiped out over an infinite time line(setting aside heat death of course).

I get that Isaac is a science popularizer and he needs to explain things in broader terms with out going into every exception. This isn't an obscure exception but rather one that has a lot practical application. It is a folly for a degenerate gambler to decide to stop his self destructive behavior only after he breaks-even regardless if the gambler has an infinite amount of time and money to pursue such a quest. There is some positive probability X that earth will be ravaged by nuclear war. If we can't get the probability of X to be 0 is humanity doomed or do we have some hope? Well if we continue to decrease X we have a chance....there is hope. Because there is hope, we've got to try to continuously reduce X.

done! the latest episode added a caveat to the claim. Should get your hopes up as high as expected as well :)
www.youtube.com/watch?v=cCzpYxedIe0&t=14m35s