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Expected value gives you the "If I look at 5000 shower-takings with a 1/1000 probability of a shower-taking resulting in death, I expect to see 5 deaths on average". Of course when applying it to a single lifetime, the correct thing to consider is "If I take showers until I have taken 5000 showers or died, what is the expected number of shower-deaths I will incur", which is .9972.

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