IM3.7.3 TWT and CP
Students at Rufus King High School were discussing some of the challenges of finding space for athletic teams to practice after school. Part of the problem, according to Kristin, is that female students are more likely to be involved in an after-school athletics program than male students. However, the athletic director assigns the available facilities as if male students are more likely to be involved. Before suggesting changes to the assignments, the students decided to investigate.[br]Suppose the following information is known about Rufus King High School: 40% of students are involved in one or more of the after-school athletics programs offered at the school. It is also known that 58% of the school’s students are female. The students decide to construct a two-way [b]relative frequency[/b] table, entering percentages (as decimals) instead of numbers of students, to organize the data.
1. What cell in the table represents 100% of the students at Rufus King High School? [br][br]2. What cells in the table can be filled based on the information given about the student population? Place these values in the appropriate cells of the table based on this information.[br][br]3. Based only on the cells you completed in Exercise 2, which of the following probabilities can be calculated, and which cannot be calculated? Calculate the probability if it can be calculated. If it cannot be calculated, indicate why.[br]a. The probability that a randomly selected student is female[br][br]b. The probability that a randomly selected student participates in an after-school athletics program[br][br]c. The probability that a randomly selected student who does not participate in an after-school athletics program is male[br][br]d. The probability that a randomly selected male student participates in an after-school athletics program[br][br]4. The athletic director indicated that 23.2% of the students at Rufus King are female and participate in after-school athletics programs. Based on this information, complete the table above.
Academic language and Mathematical Notation
Let A represent the event “a randomly selected student is female.”Let B represent the event “a randomly selected student participates in an after-school athletics program.”[br]Use the table above to show what should be in place of numbers 5-10.
Calculating Probabilities
11. Using your table, find the probabilities of the following.[br]a. A[br]b. B[br]c. A'[br]d. B'[br]e. [math]A\cup B[/math][br]f. [math]A\cap B[/math]
Conditional Probabilities
Conditional probabilities can be calculated by focusing on specific rows or columns in a TWT. The probability of A being true [i]given[b] [/b]the condition that B is also true[/i] is written [math]Pr\left(A|B\right)[/math]. Write the following using mathematical notation, and calculate the probability.[br][br]12. What is the probability that if a randomly selected student is female, she participates in the after-school athletic program?[br]13. What is the probability that if a randomly selected student is female, she does not participate in after-school athletics?[br]14. A randomly selected student is male. What is the probability he participates in an after-school athletics program?
Making decisions based on probability
15. Based on your answers to 12-14, do you think that female students are more likely to be involved in after-school athletic programs? Explain your answer.