In this activity, students use the context of a simple game to ask their own question involving independence of events Then they collect data to answer their question.

Students must model mathematically to determine what information they need to answer their question (MP4). They also construct a viable argument to answer their question (MP3).

The purpose of this discussion is for students to communicate how they used mathematics to justify their findings.

• “Which events did you choose to compare?” (I chose to compare the event of me displaying rock and winning.)
• “How did you determine whether or not the two events are independent?” (I computed and compared that to . Because the probabilities were not close to the same, I think the events might be dependent.)
• “What was challenging or interesting about this activity?” (I thought it was interesting because we collected our own data and actually used probability to answer questions.)

In this activity, students apply conditional probability to make sense of a nonintuitive statistical problem. Students play a game of drawing a card from a bag and observing one side of the card. Students then must guess which card they drew based on what they see.

Some students may have difficulty figuring out the answer to question 4. Prompt students to think about the total number of sides rather than the number of cards. Emphasize that the number of sides are the sample space, not the number of cards.

The purpose of this discussion is for students to communicate their reasoning with each other.

Poll the class to collect information about what was shown and which card it turned out to be.

Students should notice that two of the positions (top right and bottom left) must contain zeros.

Here are some questions for discussion.

• “How did you figure out the answer to question 4?” (I figured out that it was because . We know (there are 6 sides that could be showing and 2 of them are Xs from the card with an X on both sides) and (there are 6 sides that could be showing, and 3 of them have an X). Then we can find which is .)
• “Do you think that playing the game 100 times in question 2 would have helped you arrive at the answer to question 4? Explain your reasoning.” (I think it might have made me feel more confident in my answer to question 4. Because my group played the game only 12 times, it really did not give me much insight. I wonder what would have happened if we played it 100 times. Maybe we could combine the data from the whole class to get a better idea of what would happen in the long run.)