4. Probability
Counting
2:15 minutesProblem 3.T.4d
Textbook Question
4. The table on the left shows the secondary school student enrollment levels (in thousands by grade) in Oklahoma and Texas schools in a recent year. (Source: U.S. Nation
for Education Statistics)
A student in one of the indicated grades and states is randomly selected. Find the probability of selecting a student who
d. is enrolled in Texas, given that the student is in twelfth grade.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the conditional probability of selecting a student enrolled in Texas, given that the student is in twelfth grade. Conditional probability is calculated using the formula P(A|B) = P(A ∩ B) / P(B), where A is the event 'student is enrolled in Texas' and B is the event 'student is in twelfth grade'.
Step 2: Identify the relevant data from the table. From the table, the number of twelfth-grade students in Texas is 353.3 (in thousands), and the total number of twelfth-grade students across both states is 397.4 (in thousands).
Step 3: Calculate P(B), the probability of a student being in twelfth grade. This is the ratio of the total number of twelfth-grade students to the total number of students across all grades: P(B) = Total twelfth-grade students / Total students = 397.4 / 1758.6.
Step 4: Calculate P(A ∩ B), the probability of a student being enrolled in Texas and in twelfth grade. This is the ratio of the number of twelfth-grade students in Texas to the total number of students across all grades: P(A ∩ B) = Twelfth-grade students in Texas / Total students = 353.3 / 1758.6.
Step 5: Use the conditional probability formula to find P(A|B). Substitute the values from Steps 3 and 4 into the formula: P(A|B) = P(A ∩ B) / P(B).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conditional Probability
Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In this context, it involves calculating the probability of selecting a student from Texas, given that the student is already known to be in twelfth grade. This is expressed mathematically as P(Texas | Twelfth Grade) and is calculated using the formula P(A | B) = P(A and B) / P(B).
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Total Probability
Total probability is the sum of the probabilities of all possible outcomes of a random variable. In this scenario, it involves understanding the total number of twelfth-grade students enrolled in both Oklahoma and Texas, which is necessary to determine the denominator in the conditional probability calculation. The total number of twelfth graders is crucial for finding the probability of selecting a student from Texas.
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Data Interpretation
Data interpretation involves analyzing and making sense of data presented in tables or graphs. In this case, it requires extracting relevant enrollment figures from the provided table to compute the required probabilities. Understanding how to read and interpret the data accurately is essential for solving the problem and ensuring that the calculations are based on the correct figures.
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