British geologists have discovered that Mt. Etna erupts about once every 25 years.
Making the simplifying assumption that volcanic activity is independently distributed on a year-by-year basis and occurs with constant probability how many years would it take to be sure that Mt. Etna has erupted?
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Let p n be the probability that Mt. Etna doesn't erupt in n years.
As the probability of eruption is independent for each year, we can calculate p n using the rule of product. Each year has a probability of 0 . 9 6 of no eruption, so p n = ( 0 . 9 6 ) n .
The probability that Mt. Etna erupts at least once in n years is 1 − p n . So we need 1 − p n > 0 . 5 ⟹ p n < 0 . 5 .
So ( 0 . 9 6 ) n < 0 . 5 . The smallest value of n for which this is true is n = 1 7 .