The purpose of this discussion is to make the connection between a two-way table, relative frequencies, and estimated probabilities. Here are some questions for discussion.

• “Using the data from this activity, how would you draw a similar table showing the estimated probabilities for each cell in the table?” (Divide each cell by the total number of rolls.)
• “Which roll had the highest estimated probability?” (My group rolled 60 times and 3,4 showed up 7 times.)
• “Which roll had the lowest estimated probability?” (The lowest estimated probability for my groups was 2,2 and 6,1. Each of these appeared only once. The estimated probability for each was because my group rolled 70 times.)
• “How are relative frequencies and estimated probability related?” (The relative frequency for the total number rolls can be used to estimate the probability.)

Some students may struggle finding the probability for the questions in which a person is selected from only a subgroup of the population. Prompt students to look at the table of the students in a school district and ask, "How many students are in grade 9?" (4,565) and then ask, "We ask all the students in grade 9 to come into the gymnasium, then select from only the people in the room. What is the probability that the student we select comes to school in a car?” () Emphasize that the denominator in the probability is the total number of students in grade 9 (4,565) and the numerator is the number of students in that subgroup who take a car to school (1,141).

The goal of this discussion is to make sure that students understand how to estimate probabilities using information in a two-way table.

Here are some questions for discussion.

• “How do you use a two-way table to estimate probabilities?” (You look at the cell determined by the question being asked and divide it by the total for the group or subgroup involved in the question.)
• “How do you know when to use the total for the whole table, or a total for a row or column to estimate the probability?” (You have to pay close attention to how the question is worded. For example, for the second table, if it is a high school student selected at random, then you would use the total for the whole table. On the other hand, if it is a student from one grade being selected at random, then you would use the total for just that grade.)
• “Did any of the probability questions stand out to you as more difficult? Explain your reasoning.” (I found that the questions that used row or column totals were more difficult because I could not just use the total for the table by default. I really had to think about what the question was asking for.)