In Evan's history class, 10 out of 100 key terms will be...

In Evan's history class, 10 out of 100 key terms will be randomly selected to appear on the final exam; Evan must then choose 7 of those 10 to define. Since he knows the format of the exam in advance, Evan is trying to decide how many key terms he should study. (a) Suppose that Evan decides to study $s$ key terms, where $s$ is an integer between 0 and $100 .$ Let $X$ be the number of key terms appearing on the exam that he has studied. What is the distribution of $X ?$ Give the name and parameters, in terms of $s$. (b) Using $R$ or other software, calculate the probability that Evan knows at least 7 of the 10 key terms that appear on the exam, assuming that he studies $s=75$ key terms.

In Evan's history class, 10 out of 100 key terms will be randomly selected to appear on the final exam; Evan must then choose 7 of those 10 to define. Since he knows the format of the exam in advance, Evan is trying to decide how many key terms he should study.
(a) Suppose that Evan decides to study $s$ key terms, where $s$ is an integer between 0 and $100 .$ Let $X$ be the number of key terms appearing on the exam that he has studied. What is the distribution of $X ?$ Give the name and parameters, in terms of $s$.
(b) Using $R$ or other software, calculate the probability that Evan knows at least 7 of the 10 key terms that appear on the exam, assuming that he studies $s=75$ key terms.

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Key Concepts

Hypergeometric Distribution

This is a probability distribution used to model the number of successes in a fixed-size sample drawn without replacement from a finite population. In the context of the problem, the total set of key terms is the population, the studied terms are considered ‘successes’ in the population, and the key terms chosen for the exam form the sample. The distribution is characterized by parameters N (the population size), K (the number of successes in the population, represented by s key terms studied), and n (the sample size, which is 10 key terms on the exam). Its probability mass function is given by P(X=k) = [C(s,k) * C(100-s, 10-k)] / C(100,10).

Cumulative Probability Calculation

This involves summing the probabilities of all outcomes from a given threshold to the maximum value. In this case, the calculation of P(X>=7) means adding up the probabilities for X=7, 8, 9, and 10. Such cumulative probabilities are often computed using statistical software like R, which can efficiently handle combinatorial calculations and summing multiple probabilities to arrive at the final likelihood value.

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