Lesson 11

The mathematical purpose of this activity is for students to collect data, look for patterns in data, and to use probability to decide if events are independent.

Arrange students in groups of two. Invite two students to demonstrate the game “Rock, Paper, Scissors.”

Students should be encouraged to think of their own hypotheses to study, but here are some examples of things for students who struggle to think of something.

• One of the options (for example, rock) tends to result in a win for me.
• The person who won last tends to win again more often.
• Playing the same option two games in a row tends to win more often on the second time.

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 paper and winning.)
• “How did you determine whether or not the two events are independent?” (I computed \(P(\text{win | played rock})\) and compared that to \(P(\text{win})\). Since 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.)

The mathematical purpose of this activity is for students to apply conditional probability in the context of a problem. It is important that students use the ruler to draw the X on each card so that slight differences in the way the Xs are drawn are not recognizable.

Arrange students in groups of 2. Give 3 cards and one ruler to each group.

Demonstrate the game by playing one round with the students.

Tell students, "The first card will be blank on both sides, the second card will have an X on one side, and the third card will have an X on both sides. It is important to use the ruler when drawing the Xs on the cards since you should not be able to guess the card based on small differences in how the X is drawn." 

Place all three cards in the bag. Remove one and hold it against the wall so that only one side can be seen by the students. Ask them which card they think this card is based on what they are seeing. Is it the blank card, the card with one X on one side, or the card with an X on both sides?

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 on 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 \(\frac{2}{3}\) since \(P(\text{X on both sides | saw an X}) = \frac{P(\text{X on both sides and saw an X})}{P(\text{saw an X})}\). We know \(P(\text{the card had an X on both sides and the side showing had an X on it}) = \frac{2}{6}\) (there are 6 sides that could be showing and 2 of them are Xs from the card with an X on both sides) and \(P(\text{the side showing had an X on it}) = \frac{3}{6}\) (there are 6 sides that could be showing and 3 of them have an X). Then we can find \(P(\text{the card had an X on both sides | the side showing had an X on it}) = \frac{ \left( \frac{2}{6} \right)}{\left( \frac{3}{6} \right)}\) which is \(\frac{2}{3}\).
• “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. Since my group only played the game 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 ideas of what would happen in the long run.)