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In general, harmful mutations will die out. In order to spread to a significant proportion of the population, yes, a random mutation has to be lucky. It has to random-walk in a very rare way, and it is still more likely than not going to hit the gambler's ruin and be eventually eliminated from the population, even if it first spreads to 99% of said population (an extremely unlikely event).
But the thing about random-walking is that it is random. One wouldn't bet on a given harmful mutation spreading fast (not if one wanted to win the bet)... but if there are a million harmful mutations, then one of them could reasonably be expected to have one-in-a-million luck.
I think we're pretty much on the same page. But have you actually calculated the odds? One in a million is no big deal. Twenty half-chances.
I must say I haven't, and I don't know how to (especially since it's all screwed up by genes moving around and getting passed on together, and I don't understand the first thing about all that). But it feels more like 'thermodynamic entropy' than 'winning the lottery'.
Also remember that nothing is perfectly neutral. Even the banana man might get fed banana-cake by a dastardly enemy.
No, I haven't actually calculated the odds. I wouldn't really have much of an idea how. (I could probably work it out on a basis of - if a gene has x% chance of preventing descendants as compared to not having that gene and a y% chance of being passed on to any descendants - and then do some overly-simplified calculations from the values of x and y - but I haven't, yet.)
True, but his problem there isn't the banana gene. His problem there is that he has a dastardly enemy. If he didn't have the banana gene, the dastardly enemy could simply feed him arsenic cake instead, or just shoot him.