Five-Minute Check (over Lesson 12–7) Mathematical Practices Then/Now New Vocabulary Example 1: Real-World Example: Make a Two-Way Frequency Table Example 2: Use Marginal and Joint Frequencies Example 3: Determine Relative Frequencies Example 4: Decide Whether Events Are Independent Example 5: Find a Conditional Probability Lesson Menu

Determine whether the event is independent or dependent Determine whether the event is independent or dependent. Samson ate a piece of fruit randomly from a basket that contained apples, bananas, and pears. Then Susan ate a second piece from the basket. A. independent B. dependent 5-Minute Check 1

Determine whether the event is independent or dependent Determine whether the event is independent or dependent. Kimra received a passing score on the mathematics portion of her state graduation test. A week later, she received a passing score on the reading portion of the test. A. independent B. dependent 5-Minute Check 2

A spinner with 4 congruent sectors labeled 1–4 is spun A spinner with 4 congruent sectors labeled 1–4 is spun. Then a die is rolled. What is the probability of getting even numbers on both events? A. 1 B. C. D. 5-Minute Check 3

Two representatives will be randomly chosen from a class of 20 students. What is the probability that Janet will be selected first and Erica will be selected second? A. B. C. D. 5-Minute Check 4

Mathematical Practices 2 Reason abstractly and quantitatively. 4 Model with mathematics. Content Standards S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. MP

You found conditional probabilities. Decide whether events are independent by using two-way frequency tables. Approximate conditional probabilities by using two-way frequency tables. Then/Now

two-way frequency table marginal frequency joint frequency relative frequency New Vocabulary

Make a Two-Way Frequency Table Valerie asks a random sample of 70 science teachers and math teachers whether they have been to the town’s planetarium. She finds that 25 science teachers have been to the planetarium and 3 have not, while 20 math teachers have been to the planetarium and 22 have not. Make a two-way frequency table to organize the data. Make a table with 4 columns and 4 rows. Label the second column for teachers who have been to the planetarium. Label the third column for teachers who have not been to the planetarium. Label the fourth column for the totals. Real-World Example 1

Has been to Planetarium Has not been to Planetarium Make a Two-Way Frequency Table Label the second row science teachers. Label the third row math teachers. Label the fourth row totals. Use the information given to fill in the table. Has been to Planetarium Has not been to Planetarium Totals Science 25 3 28 Math 20 22 42 45 70 Answer: Real-World Example 1

Use Marginal and Joint Frequencies Deniqua collected data from a random sample of customers at two restaurants to find out if they paid by cash or credit card. The data are shown in the two-way frequency table. Answer each question and tell whether you are using a marginal frequency or joint frequency. Example 2

A. How many customers at Donna’s Diner paid by credit card? Use Marginal and Joint Frequencies A. How many customers at Donna’s Diner paid by credit card? Look at the table. Find the intersection of Donna’s Diner and credit card. What number is in the cell? 11 Is the frequency marginal or joint? Joint Answer: 11; joint frequency Example 2A

B. How many customers paid by cash? Use Marginal and Joint Frequencies B. How many customers paid by cash? Look at the table. Find the intersection of totals and cash. What number is in the cell? 28 Is the frequency marginal or joint? Marginal Answer: 28; marginal frequency Example 2A

Convert the below table to a table of relative frequencies. Determine Relative Frequencies Convert the below table to a table of relative frequencies. Example 3

Simplify and convert to percent. Determine Relative Frequencies To find the relative frequencies divide the value in each cell by the total numbers of respondents. Joe’s Place Donna’s Diner Totals Cash 20 ÷ 50 8 ÷ 50 28 ÷ 50 Credit Cards 11 ÷ 50 22 ÷ 50 31 ÷ 50 19 ÷ 50 50 ÷ 50 Simplify and convert to percent. Joe’s Place Donna’s Diner Totals Cash 0.40 0.16 0.56 Credit Cards 0.22 0.44 0.62 0.38 1 Example 3

Determine Relative Frequencies Answer: Example 3

Decide Whether Events Are Independent Use the below relative frequency table to determine whether paying by cash or credit card is independent of the restaurant. Explain. Joe’s Place Donna’s Diner Totals Cash 40% 16% 56% Credit Cards 22% 44% 62% 38% 100% Example 4

Are the expected and actual joint relative frequencies the same? No Decide Whether Events Are Independent Are the expected and actual joint relative frequencies the same? No Is paying by cash or credit card independent of the restaurant? No Answer: No; the expected and actual joint relative frequencies are not the same. Example 4

Find a Conditional Probability Use the below relative frequency table to find the probability that a customer pays by cash, given that he or she eats at Joe’s Place. Joe’s Place Donna’s Diner Totals Cash 40% 16% 56% Credit Cards 22% 44% 62% 38% 100% Example 5

Find the intersection of Joe’s Place and cash. Find a Conditional Probability Find the intersection of Joe’s Place and cash. What number is in the cell? 40% Find the intersection of Joe’s Place and totals. What number is in the cell? 62% Divide to find the probability. 40 ÷ 62 ≈ .64516 Convert to a percentage. 64.5% Answer: 64.5% Example 5