# Lesson 11

Probabilities in Games

## 11.1: Rock, Paper, Scissors (20 minutes)

### Optional activity

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.

### Launch

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.

### Student Facing

There is a classic game called “Rock, Paper, Scissors.” Two people play by counting to 3 together, then making a hand gesture to resemble paper (hand flat, palm down), rock (fist), or scissors (two fingers extended).

- When paper and rock are shown, paper wins the round.
- When paper and scissors are shown, scissors wins the round.
- When rock and scissors are shown, rock wins the round.
- When both players show the same thing, the round is a tie.

Find a partner and play the game with them 10 times in a row. Record the number of times you have played the game, the name of your opponent, what each person shows in each round, and who is the winner. Find another partner and play another 10 times in a row.

Is the event “win the round” dependent on another event? Explain your reasoning.

Choose an event that you think might influence the probability of winning, then analyze the data using probability to determine whether the event you chose to study is independent of winning. Provide evidence to support your claim.

### Student Response

### Activity Synthesis

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.)

*Speaking, Representing: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.

*Design Principle(s): Support sense-making*

## 11.2: Guess Which Card (20 minutes)

### Optional activity

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.

### Launch

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?

*Engagement: Provide Access by Recruiting Interest.*Begin with a small-group or whole-class demonstration of how to play the game. Check for understanding by inviting students to rephrase directions in their own words.

*Supports accessibility for: Memory; Conceptual processing*

### Student Facing

Your teacher will give you 3 index cards.

- Leave one card blank.
- Use a ruler to draw lines connecting opposite corners to make an X on one side of the second card.
- Use a ruler to draw lines connecting opposite corners to make an X on both sides of the third card.
- Put all three cards in the bag.

One partner will remove a card from the bag and place it on the desk immediately so that only one side of the card can be seen. The goal is to guess correctly which card is on the desk: the blank card, the card with an X only on one side, or the card with an X on both sides.

- Noah is playing the game and is looking at a card that shows an X. He says, “I have a fifty-fifty chance of correctly guessing which card it is.” Do you agree with Noah? Explain your reasoning.
- Play the game many times with your partner, taking turns for who takes a card out of the bag. Record whether the side that shows has an X or is blank, which card you guess, and which card it actually is when you check. Continue to play until your teacher tells you to move on.
- Use the results from your games to estimate \(P(\text{the card had an X on both sides | the side showing had an X on it})\). Explain or show your reasoning.
- Since all of the outcomes are known, find the actual probability \(P(\text{the card had an X on both sides | the side showing had an X on it})\). Explain or show your reasoning.

### Student Response

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

both sides blank | one X | two Xs | |
---|---|---|---|

blank side shown | |||

X shown |

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.)

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion.

- “What did you find important, challenging, or otherwise notable about today’s lesson?” (It was really interesting to collect data to answer a question. I really felt like I was able to better understand the concept of probability.)
- “What mathematical work from this unit did you use in today’s activity?” (I used probability, conditional probability, the addition rule, and relative frequencies.)
- “How might the mathematical tools that you have learned in this unit help you outside of mathematics class?” (I think that when I see data that compares data using percentages in the news that I will really have to stop and think about the different factors that might account for any differences. I wonder why I never hear about independent and dependent events when I look at data in the news. I think it would be really interesting if the graphs or tables used by the news had to explain them using the concept of probability.)