Predicting Volcano Activity

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 4 % , 4 \%, how many years would it take to be > 50 % > 50 \% sure that Mt. Etna has erupted?

15 years 13 years 17 years 19 years

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1 solution

Julian Yu
Dec 22, 2018

Let p n p_n be the probability that Mt. Etna doesn't erupt in n n years.

As the probability of eruption is independent for each year, we can calculate p n p_n using the rule of product. Each year has a probability of 0.96 0.96 of no eruption, so p n = ( 0.96 ) n p_n=(0.96)^n .

The probability that Mt. Etna erupts at least once in n n years is 1 p n 1-p_n . So we need 1 p n > 0.5 p n < 0.5. 1-p_n>0.5 \implies p_n<0.5.

So ( 0.96 ) n < 0.5. (0.96)^n<0.5. The smallest value of n n for which this is true is n = 17 n=17 .

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