Students in grades 3 through 8 in New York State are required to take a state mathematics exam. To meet the state's proficiency standards, a student must demonstrate an understanding of the mathematics expected at his or her grade level. The following data show the number of students tested in the New York City school system for grades 3 through 8 and the number who met and did not meet the proficiency standards on the exam (New York City Department of Education website, January 16, 2010).

a. Develop a joint probability table for these data.
b. What are the marginal probabilities? What do they tell about the probabilities of a student meeting or not meeting the proficiency standards on the exam?
c. If a randomly selected student is a third grader, what is the probability that the student met the proficiency standards? If the student is a fourth grader, what is the probability that the student met the proficiency standards?
d. If a randomly selected student is known to have met the proficiency standards on the exam, what is the probability that the student is a third grader? What is the probability if the student is a fourth grader?

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Met Proficiency Standards? Yes 47,401 35,020 36,062 36,361 40,945 40,720 No 23,975 34,740 33,540 32,929 29,768 31,931 Grade 4