How does the Physics work for the Quantum Suicide thought experiment?

On page 5 of this paper written by Max Tegmark, Tegmark discusses a thought experiment called 'Quantum Suicide'. As far as I understand it, this experiment was created to show the experimental difference between the Copenhagen interpretation and the Many Worlds Interpretation of Quantum Mechanics.

I'm really not sure how the physics works out here. As far as I understand it, the Copenhagen Interpretation assures us that after each measurement, the wave-function collapses for the particle, and asserts that the experimenter will either be dead or alive after each measurement. So the experimenter, as long as she isn't incredibly lucky, should be dead within a couple of measurements. This all works out in my head very nicely, and makes a lot of sense. However, somehow when we are dealing with the Many Worlds Interpretation, "the observer will hear click with 100% certainty". Somehow, the experimenter is guaranteed to survive. At first, I thought this concept is just a parallel of the Anthropic Principle, but on further consideration that doesn't seem to be the case. The Wikipedia Article for this thought experiment states:

However, if the many-worlds interpretation is true, a superposition of the live experimenter necessarily exists, regardless of how many iterations or how improbable the outcome. Barring life after death, it is not possible for the experimenter to experience having been killed, thus the only possible experience is one of having survived every iteration.

Sure, there is a possibility that the experimenter survives. But isn't that the same with the Copenhagen interpretation? Basically, I am very confused, and any help to understand this thought experiment, and its implications (or the maths, because I don't understand equation 4), would be great.