California is hit every year by approximately 500 earthquakes that are large enough to be felt. However, those of destructive magnitude occur, on average, once a year. Find a) Probability that at least 3 months elapse before the first earthquake of destructive magnitude occurs. b) Probability that at least 7 months elapsed before the first earthquake of destructive magnitude occurs knowing that 3 months have already elapsed. Expert Answer

Answer:a) The probability that at least 3 months elapse before the first earthquake of destructive magnitude occurs is P=0.7788b) The probability that at least 7 months elapsed before the first earthquake of destructive magnitude occurs knowing that 3 months have already elapsed is P=0.7165Step-by-step explanation:Tha most appropiate distribution to model the probability of this events is the exponential distribution.The cumulative distribution function of the exponential distribution is given by:[tex]P(t<x;\lambda)=1-e^{-\lambda t}[/tex]The destructive earthquakes happen in average once a year. This can be expressed by the parameter λ=1/year.We can express the probability of having a 3 month period (t=3/12=0.25) without destructive earthquakes as:[tex]P(t>0.25)=1-P(t<0.25)=1-(1-e^{-1*0.25})=e^{-1*0.25}=0.7788[/tex]Applying the memory-less property of the exponential distribution, in which the past events don't affect the future probabilities, the probability of having at least 7 months (t=0.58)  elapsed before the first earthquake given that 3 months have already elapsed, is the same as the probability of having 4 months elapsed before an earthquake.[tex]P(t>0.58)/P(t>0.25)=P(t>0.33)[/tex][tex]P(t>0.33)=1-P(t<0.33)=e^{-1*0.33}=0.7165[/tex]