# Discussion: Probability and Probability Distributions

## Discussion: Probability and Probability Distributions

### Discussion: Probability and Probability Distributions

You are the healthcare administration leader for a health services organization and are interested in achieving a standard, whereby 90% of all patients are screened within the initial 15 minutes of arriving for a family practice appointment Discussion: Probability and Probability Distributions.

Part 1

In a random sample of 30 patients, you find that 25 were screened within 15 minutes. The probability that this event (or one more extreme) would occur, might be modeled as a binomial with the following probability statement:

P(X≤25 | N=30, p=0.9)

To solve, you use =binom.dist(25, 30, 0.9, TRUE) in Excel and find that you would expect 25 or fewer screenings in 30 trials when the success rate should be 0.9 about 17.5% of the time.

Review the resources and consider how one of the distributions presented might be useful for healthcare administration leaders. Describe in 2 or 3 paragraphs an example of how one of the distributions presented might be used in your health services organization or one with which you are familiar Discussion: Probability and Probability Distributions.

Then, generate a representative probability statement based on the scenario and solve using fictitious data. Be specific in your probability statement.

Part 2:

Your company is running a Medicare audit on Sleaze Hospital. Because Sleaze has a history of overbilling, the focus of your audit is on checking whether the billing amounts are correct. Assume that each invoice is for too high an amount with probability 0.06 and for too low an amount with probability 0.01 (so that the probability of a correct billing is 0.93) Discussion: Probability and Probability Distributions. Also, assume that the outcome for any invoice is probabilistically independent of the outcomes for other invoices.

For this Assignment, reflect on the case presented. Think about what strategies you might use to calculate associated probabilities for Sleaze Hospital, and then address the series of questions.

### The Assignment: (3–5 pages)

• If you randomly sample 200 of Sleaze’s invoices, what is the probability that you will find at least 15 invoices that overcharge the customer? What is the probability you won’t find any that undercharge the customer?
• Find an integer, k, such that the probability is at least 0.99 that you will find at least k invoices that overcharge the customer. (Hint: Use trial and error with the BINOMDIST function to find k.)
• Suppose that when Sleaze overcharges Medicare, the distribution of the amount overcharged (expressed as a percentage of the correct billing amount) is normally distributed with mean 15% and standard deviation 4%.
• What percentage of overbilled invoices are at least 10% more than the legal billing amount?
• What percentage of all invoices are at least 10% more than the legal billing amount?
• If your auditing company samples 200 randomly chosen invoices, what is the probability that it will find at least five where Medicare was overcharged by at least 10%?
• Discussion: Probability and Probability Distributions

### References for Discussion: Probability and Probability Distributions:

Albright, S. C., & Winston, W. L. (2015). Business analytics: Data analysis and decision making (5th ed.). Stamford, CT: Cengage Learning.

• Chapter 4, “Probability and Probability Distributions” (pp. 139–165)
• Chapter 5, “Normal, Binomial, Poisson, and Exponential Distributions” (pp. 166–221)
• Discussion: Probability and Probability Distributions

Fulton, L. V., Mendez, F. A., Bastian, N. D., & Musal, R. M. (2012). Confusion between odds and probability, a pandemic? Journal of Statistics Education, 20(3), 1–20.