a) The probability of exactly three eruptions in the next 200 years is 0.140373, and b) The probability of the next eruption occurring between 25 and 50 years is 0.8698.
Given,
average number volcanic eruptions in every 100 years = 2.5
rate of success for 100 years =[tex]\frac{2.5}{100}=0.025[/tex]
a)probability of exactly three eruptions in the next 200 years
then the average volcanic eruptions in every 200 years will be 200*0.025=5
[tex]probability=\frac{\lambda^x}{x!}e^{-\lambda}[/tex]
where, x= expected number of occurrences
λ=number of occurrences during interval
[tex]P(x=3)=\frac{5^3}{3!}e^{-5}=0.140373[/tex]
b) probability the next eruption occurs between 25 and 50 years
It means we need to find sum of probabilities for no eruption for first 25 years and one eruption for next 25 years.
then,
average volcanic eruptions in every 25 years will be 25*0.025=0.625
probability=P(x=0)+P(x=1)
[tex]=\frac{0.625^0}{0!}e^{-0.625}+\frac{0.625^1}{1!}e^{-0.625}\\\\=0.53526+0.334538\\\\=0.8698[/tex]
Thus, The probability of exactly three eruptions in the next 200 years is 0.140373, and the probability of the next eruption occurring between 25 and 50 years is 0.8698.
To learn more poisson distribution refer here
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