In Section 2.3, 20 patients for Example 2.5 were randomly selected from a population of 25,000 .

a. Explain why the probability of choosing a random sample of 20 from 25,000 is

\[\frac{1}{\left(\begin{array}{c}25000 \\20\end{array}\right)}\]

b. Use $\mathrm{R}$ to randomly select 4 patients from a total of 8 patients, and calculate the probability of choosing this random sample.

Data from Example 2.5

Data from Section 2.3

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Example 2.5 (COVID-19 Treatment Study). A 2020 National Institutes of Health (NIH) sponsored study to evaluate if a treatment for COVID-19 is effective planned to enroll 20 subjects and randomly assign 10 to the treatment and 10 to a fake treatment (placebo). Effectiveness was evaluated by the proportion of participants who died from any cause or were hospitalized during the 20-day period from and including the day of the first (confirmed) dose of study treatment [Clinicaltrial.gov, 2020]. Simulated data for this study data is shown in Table 2.2. trt records if a subject received treatment TRT or placebo PLA, die_hosp is 1 if a participant died or was hospitalized in the first 20 days after treatment, and 0 otherwise,