Consider the hospital patients in Example 2-8 . Two patients...

Consider the hospital patients in Example $2-8 .$ Two patients are selected randomly, with replacement, from the total patients at Hospital $1 .$ What is the probability mass function of the number of patients in the sample who are admitted?

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Key Concepts

Probability Mass Function (PMF)

The probability mass function provides a complete description of the distribution of a discrete random variable by specifying the probability that the variable takes each of its possible values. In the context of counting admitted patients, the PMF would assign probabilities to the possible counts (e.g., 0, 1, or 2 admitted patients) based on the underlying random process.

Sampling with Replacement

Sampling with replacement means that each time a patient is selected, the entire population is available for selection again, ensuring that each draw is independent of the others. This independence is crucial when calculating the likelihood of different outcomes since the probability for each draw remains constant.

Binomial Distribution

When sampling with replacement, if each individual selection is modeled as a Bernoulli trial (with success defined as the patient being admitted and failure as not being admitted), then the total number of successes out of a fixed number of independent trials follows a Binomial distribution. The binomial distribution provides the framework for calculating the probability of obtaining a given number of admitted patients in the sample.

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