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A company determines its earthquake insurance rates based on these assumptions:
At most one earthquake can occur in a calendar year.
The probability of an earthquake occurring in a given year is 0.05.
The occurrence of earthquakes in different years is independent.
Using these assumptions, calculate the probability of experiencing fewer than three earthquakes over a 20-year period.
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The number of earthquakes expected to impact a specific house over the next ten years follows a Poisson distribution with a mean of 4. Each earthquakes causes a financial loss that follows an exponential distribution with an average of $1000.
Losses are mutually independent and independent of the number of earthquakes.
Calculate the variance of the total financial loss caused by earthquakes over the ten-year period.
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In a study, individuals are tested for COVID-19, one at a time, until a person is found to be positive for the virus. Each patient independently has the same probability of having COVID-18. Let \(r\) represent the probability that at least four individuals are tested.
Determine the probability that at least twelve individuals are tested, given that at least four individuals are tested.
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In a classroom of 25 students, 20 are undergraduates, and five are postgraduates.
Two of the 25 students are randomly chosen without replacement.
Calculate the probability that exactly one of the two selected students is an undergraduate.
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In a classroom of 25 students, 20 are undergraduates, and five are postgraduates.
Two of the 25 students are randomly chosen with replacement.
Calculate the probability that exactly one of the two selected students is an undergraduate.
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Let \(X_1, X_2, ..., X_{100}\) be independent identically distributed random variables such that \(P[X=0]=P[X=2]=0.5\). Let \(T=X_1+X+2+...+X_{100}\). Calculate the approximate value of \(P[T >115]\).
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For a person with the disease, the test will give a “negative” result (disease not detected) with a probability of 0.10.
For a person without the disease, the test will give a “positive” result (incorrectly detecting the disease) with a probability of 0.20.
20% of people being tested actually have the disease.
We want to calculate the probability that a person has the disease, given that their test result is “positive” (disease detected).
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A home owner purchases an insurance policy to offset costs associated with excessive amounts of flood. For every full 1 inches of flood in excess of 4 inches, the insurer pays the pwner Rs. 2000 up to a policy maximum of Rs. 5000. The following table shows a probability function for the random variable X of flood level, in inches, at the home.
Calculate the standard deviation of the amount paid under the policy.
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The monthly income that an sales agent earns is modeled by a random variable X with probability density function
[in-class]
Calculate the probability that the income the agent earns in a month is within 0.5 standard deviations of E(X).
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