Question

HELP!! 30 points pls help

Students at a virtual school are allowed to register for one math class each year. The number of students signing up for various math classes for the next year are given in the following table.


Geometry Algebra II Probability and Statistics Pre-Calculus Total
10th Grade 150 75 25 5 255
11th Grade 50 100 75 20 245
12th Grade 10 50 100 65 225
Total 210 225 200 90 725

Part A: What is the P(student takes Algebra II)? (5 points)

Part B: What is the P(10th Grade ∪ Algebra II)? (5 points)

Part C: What is the P(Geometry|10th Grade)? (5 points)

Part D: Consider the events "A student takes Probability and Statistics" and "A student is a 10th grader." Are these events independent? Justify your answer mathematically. (5 points)

Answer 1

Part A: The P(student takes Algebra II) is 225/725, or 0.31.

Part B: The P(10th Grade ∪ Algebra II) is 175/725, or 0.24.

Part C: The P(Geometry|10th Grade) is 150/255, or 0.59.

Part D: The events "A student takes Probability and Statistics" and "A student is a 10th grader" are not independent. Mathematically, this can be determined by calculating the probability of P(A ∩ B) = P(A) * P(B), which in this case would be 0.18. If these two events were independent, P(A ∩ B) would be equal to P(A) * P(B), which is not the case here.

Answer 2

Part A:

P(student takes Algebra II) = 225/725 = 0.31 (rounded to two decimal places)

Part B:

P(10th Grade ∪ Algebra II) = P(10th Grade) + P(Algebra II) - P(10th Grade ∩ Algebra II)

From the table, we have P(10th Grade) = 255/725 and P(Algebra II) = 225/725.

To find P(10th Grade ∩ Algebra II), we look at the intersection of the 10th Grade row and the Algebra II column, which is 75. Therefore, P(10th Grade ∩ Algebra II) = 75/725.

Substituting these values into the formula, we get:

P(10th Grade ∪ Algebra II) = 255/725 + 225/725 - 75/725 = 0.66 (rounded to two decimal places)

Part C:

P(Geometry|10th Grade) = P(Geometry ∩ 10th Grade) / P(10th Grade)

From the table, we see that P(Geometry ∩ 10th Grade) = 150/725.

Also, we know that P(10th Grade) = 255/725.

Substituting these values into the formula, we get:

P(Geometry|10th Grade) = (150/725) / (255/725) = 0.59 (rounded to two decimal places)

Part D:

We need to check whether P(Probability and Statistics ∩ 10th Grade) = P(Probability and Statistics) x P(10th Grade).

From the table, we see that P(Probability and Statistics) = 200/725.

To find P(Probability and Statistics ∩ 10th Grade), we look at the intersection of the Probability and Statistics row and the 10th Grade column, which is 25. Therefore, P(Probability and Statistics ∩ 10th Grade) = 25/725.

To find P(10th Grade), we can use the value given in the table: 255/725.

Substituting these values into the formula, we get:

P(Probability and Statistics ∩ 10th Grade) = (25/725) ≠ (200/725) x (255/725) = 0.078

Since P(Probability and Statistics ∩ 10th Grade) is not equal to P(Probability and Statistics) x P(10th Grade), we can conclude that the events "A student takes Probability and Statistics" and "A student is a 10th grader" are not independent.