Question

Records show that \(30 \%\) of all patients admitted to a medical clinic fail to pay their bills and that eventually the bills are forgiven. Suppose \(n=4\) new patients represent a random selection from the large set of prospective patients served by the clinic. Find these probabilities: a. All the patients' bills will eventually have to be forgiven. b. One will have to be forgiven. c. None will have to be forgiven.

Short answer

Question: In a medical clinic, 30% of patients admitted fail to pay their bills. If a random sample of 4 new patients is selected, find the probabilities of the following scenarios: a. All the patients' bills will eventually have to be forgiven; b. One bill will have to be forgiven; c. None of the bills will have to be forgiven. Answer: a. The probability that all the patients' bills will eventually have to be forgiven is 0.0081. b. The probability that one bill will have to be forgiven is 0.4116. c. The probability that none of the bills will have to be forgiven is 0.2401.

Step-by-step solution

Identify the probability distribution and parameters

In this problem, we are dealing with a binomial distribution with \(n=4\) trials (patients) and probability of success (not paying the bill) as \(p=0.30\). Let X denote the number of patients whose bills will have to be forgiven.

Calculate the probability that all patients' bills will have to be forgiven

To find the probability of all patients' bills being forgiven, we need to find P(X=4): Using the binomial probability formula, \(P(X=k)= \binom{n}{k} p^k (1-p)^{n-k}\), we can find the required probability: \(P(X=4) = \binom{4}{4} * 0.3^4 * (1-0.3)^{4-4}= 0.3^4 = 0.0081\).

Calculate the probability that one patient's bill will have to be forgiven

To find the probability of exactly one patient's bill being forgiven, we need to find P(X=1): \(P(X=1) = \binom{4}{1} * 0.3^1 * (1-0.3)^{4-1} = 4 * 0.3 * 0.7^3 = 0.4116\)

Calculate the probability that no patient's bill will have to be forgiven

To find the probability of no patient's bills being forgiven, we need to find P(X=0): \(P(X=0) = \binom{4}{0} * 0.3^0 * (1-0.3)^{4-0} = 0.7^4 = 0.2401\) So the probabilities of the different scenarios are: a. All the patients' bills will eventually have to be forgiven: \(0.0081\) b. One bill will have to be forgiven: \(0.4116\) c. None of the bills will have to be forgiven: \(0.2401\)

Key concepts

Probability Distribution

Understanding probability distributions is crucial to grasping the nature of random events. A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It is a statistical tool that helps to model the uncertainty inherent in various processes.

For a discrete random variable, like the number of patients not paying their bills in our exercise, the probability distribution is defined by a probability mass function. This assigns a probability to each of the potential discrete outcomes. For example, the probability that exactly one patient will not pay is one such outcome.

In more practical terms, a probability distribution can be visualized as a table or graph that aligns each event or number of occurrences (like 0, 1, 2, or more patients not paying their bills) with its respective probability. This abstract concept helps us predict and understand real-world phenomena by presenting all conceivable outcomes and their corresponding likelihoods.

Binomial Probability Formula

When dealing with a situation where there are only two possible outcomes for each trial, like in our exercise where patients either pay or don’t pay their bills, we often utilize the binomial probability formula. This formula is a cornerstone of the binomial distribution, which is applicable when:

• The number of trials, denoted by 'n,' is fixed.
• Each trial has two possible outcomes: a 'success' or a 'failure'.
• The probability of success 'p' is the same for each trial.
• The trials are independent, meaning the outcome of one trial does not affect the outcome of another.

The binomial probability formula is written as:
\[P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\]
where \(P(X = k)\) is the probability of observing 'k' successes in 'n' trials, \(\binom{n}{k}\) is the binomial coefficient, 'p' is the probability of success on an individual trial, and 'k' is the number of successes we are interested in.

Applying this formula to the provided exercise about patient bills, we can precisely calculate the likelihood of various scenarios. For example, to find the probability that all four patients' bills will have to be forgiven, we can substitute the respective values into the formula, resulting in the computation that led us to the probability of 0.0081.

Random Selection Probability

The concept of random selection probability relates to the chance of a particular event occurring when individuals or items are chosen randomly from a population. In our exercise, patients are randomly selected, and the interest lies in the probability of these patients not paying their bills.

The concept aligns with basic principles of probability where each event should be independent of the others. In the context of the medical clinic scenario, the assumption is that selecting one patient is not influenced by the selection of another, which is fundamental to applying the binomial probability formula effectively.

Understanding random selection probability is vital because it mimics real-life situations where outcomes are uncertain. Calculations based on this concept are used across various disciplines, including business, healthcare, and science, to make informed decisions. By mastering this area of probability, individuals can better assess risks and predict events in uncertain situations.