Problem 2
Question
A medical study recently documented that 905 mistakes were made among the 289,411 prescriptions written during one year at a large metropolitan teaching hospital. Suppose a patient is admitted with a condition serious enough to warrant ten different prescriptions. Approximate the probability that at least one will contain an error.
Step-by-Step Solution
Verified Answer
To solve the problem, calculate the probability of having at least one error in 10 prescriptions using the binomial distribution formula, and the probability calculated in step 3 gives us the required probability.
1Step 1: Calculate the probability of a single error
Identify the probability of an error on a single prescription. This can be calculated by the ratio of total mistakes made to the total prescriptions. Therefore, \( p = \frac{905}{289411} \).
2Step 2: Use the Binomial Formula
Calculate the probability of no errors using the binomial formula: \( P(X=k) = C(n, k) * (p)^k * (1-p)^{(n-k)} \) where \( n \) is the number of prescriptions, \( p \) is the probability of a single error, and \( k \) is the number of errors we are interested in (in this case, 0). So, apply the formula for no errors: \( P(X=0) = C(10, 0) * (p)^0 * (1-p)^{(10-0)} = (1-p)^{10} \).
3Step 3: Calculate the Probability of at least one error
We need the probability of at least one error, which is equivalent to: \( 1 - P(X=0) \). Therefore, substitute the value calculated in step 2 into the equation. Thus, the probability of at least one error is \( 1 - (1-p)^{10} \).
Key Concepts
Binomial Probability DistributionMedical StatisticsProbability Calculation
Binomial Probability Distribution
In analyzing the safety of medical prescriptions, the binomial probability distribution provides valuable insights. This statistical tool is used when there are only two possible outcomes for an event (such as an error or no error on a prescription) and when the same probability of success or failure applies to each independent trial.
The formula for the binomial probability distribution is given by \( P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)} \), where \(n\) is the number of trials (prescriptions), \(k\) is the number of successes (errors) we are interested in, \(p\) is the probability of a single success (error), and \(1-p\) is the probability of a single failure (no error).
For instance, if a patient has ten prescriptions, and we wish to find the probability that at least one contains an error, we would set \(k\) to 0 (for the complementary event of no prescriptions containing an error) and subtract the result from 1 to get the probability of the opposite event. This method is particularly effective in medical statistics to estimate risks and ensure patient safety.
The formula for the binomial probability distribution is given by \( P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)} \), where \(n\) is the number of trials (prescriptions), \(k\) is the number of successes (errors) we are interested in, \(p\) is the probability of a single success (error), and \(1-p\) is the probability of a single failure (no error).
For instance, if a patient has ten prescriptions, and we wish to find the probability that at least one contains an error, we would set \(k\) to 0 (for the complementary event of no prescriptions containing an error) and subtract the result from 1 to get the probability of the opposite event. This method is particularly effective in medical statistics to estimate risks and ensure patient safety.
Medical Statistics
Medical statistics involve the application of statistical methods to medical research and practice. When dealing with prescription errors in healthcare, it's vital to use medical statistics to understand the magnitude and frequency of these mistakes. The data gathered from studies, like the one that documented 905 errors among 289,411 prescriptions, is used to calculate probabilities which inform decision-making processes to enhance patient safety.
Moreover, medical statistics help health professionals and researchers to interpret data meaningfully. In our case, finding the probability of a prescription error involves applying statistical operations to calculate a ratio, informing us of the likelihood of such events and guiding hospitals to develop strategies to minimize these occurrences. Consistently analyzing these probabilities allows for continual assessment and improvement in healthcare delivery.
Moreover, medical statistics help health professionals and researchers to interpret data meaningfully. In our case, finding the probability of a prescription error involves applying statistical operations to calculate a ratio, informing us of the likelihood of such events and guiding hospitals to develop strategies to minimize these occurrences. Consistently analyzing these probabilities allows for continual assessment and improvement in healthcare delivery.
Probability Calculation
Probability calculation is the process of determining the likelihood of a specific event occurring. This is essential in the context of prescription errors, where stakeholders can use probability to estimate risk and implement preventative measures.
To calculate the probability of at least one error in ten prescriptions, we first find the probability of no errors occurring in those ten chances. Then, we use the complement rule, which states that the probability of an event happening is equal to one minus the probability of it not happening. This approach simplifies complex probability calculations and can be applied widely, from medical scenarios to everyday decision-making. Understanding how to manipulate these probabilities is critical for students not only in grasping the steps to the solution but also in applying these techniques in real-world situations.
In our example, we would subtract the probability of no errors (computed using the binomial formula) from one to get the required probability of at least one error out of ten prescriptions. These calculations provide concrete figures that can aid in formulating policies to safeguard patient health and improve the accuracy of prescriptions.
To calculate the probability of at least one error in ten prescriptions, we first find the probability of no errors occurring in those ten chances. Then, we use the complement rule, which states that the probability of an event happening is equal to one minus the probability of it not happening. This approach simplifies complex probability calculations and can be applied widely, from medical scenarios to everyday decision-making. Understanding how to manipulate these probabilities is critical for students not only in grasping the steps to the solution but also in applying these techniques in real-world situations.
In our example, we would subtract the probability of no errors (computed using the binomial formula) from one to get the required probability of at least one error out of ten prescriptions. These calculations provide concrete figures that can aid in formulating policies to safeguard patient health and improve the accuracy of prescriptions.
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