In Exercises 17 and 18, use the table, which shows the numbers of first-time and repeat U.S. nursing students taking the National Council Licensure Examination (NCLEX-RN® exam) to pass or fail in a recent year. (Adapted from National Council Licensure Examinations)

17. Find the probability that a student took the exam for the first time, given that the student failed.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the conditional probability that a student took the exam for the first time, given that the student failed. This is a conditional probability problem, and the formula for conditional probability is P(A|B) = P(A ∩ B) / P(B).
2. Step 2: Identify the relevant data from the table. The number of students who failed the exam is 87,297 (this is P(B)). Out of these, the number of first-time students who failed is 27,522 (this is P(A ∩ B)).
3. Step 3: Write the formula for the conditional probability. Using the formula P(A|B) = P(A ∩ B) / P(B), substitute the values: P(First-time | Failed) = 27,522 / 87,297.
4. Step 4: Simplify the fraction. Divide the numerator (27,522) by the denominator (87,297) to calculate the probability. This will give the proportion of students who took the exam for the first time among those who failed.
5. Step 5: Interpret the result. The final value represents the likelihood that a student who failed the exam was taking it for the first time. Ensure the result is expressed as a decimal or percentage for clarity.

Key Concepts

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In this context, we are interested in finding the probability that a student is a first-time test taker, given that they failed the NCLEX-RN exam. This is calculated using the formula P(A|B) = P(A and B) / P(B), where A is the event of being a first-time taker and B is the event of failing.

Joint Probability

Joint probability is the probability of two events happening at the same time. In this scenario, it refers to the probability that a student both failed the exam and is a first-time test taker. This is important for calculating conditional probabilities, as it provides the numerator in the conditional probability formula.

Total Probability

Total probability is the sum of the probabilities of all possible outcomes of an event. In this case, to find the probability of failing the exam, we need to consider both first-time and repeat test takers. The total number of failures is used as the denominator in the conditional probability formula, allowing us to determine the likelihood of a student being a first-time taker among those who failed.