This lesson is based on the *MTLT* article, *“**Frustration: Analyzing a Card Game with Probability* " by Josephine Derrick, Joe Champion, and Ramey Uriarte. This lesson engages students in the joy of mathematical inquiry through a game, while building number sense, understanding of uncertainty, statistical reasoning, and discourse skills. Students will explore the ideas of experimental and conditional probability through the card game, *Frustration*.

Presentation Slides

**Warm-Up:**

Introduce the name of the game: Frustration, a card game used to analyze probability. Open the presentation slides and project them for the whole class to see. Have students get a pencil and paper, then discuss the goals and essential question.

Break down the goals to ensure a rough understanding, such as discussing what probability means. Then, review the makings of a deck of cards to ensure all students are familiar with the ranks (13) and the suits (4).

Directions for the game can be found on slide 5 of the presentation slides. Demonstrate the game with four cards at the front of the room. As you play the game, create a table on the whiteboard that matches the top two rows of the handout, and fill it in as you play to model what students will be doing. As you play, ask students to predict whether you will win or lose the next turn on the basis of the cards that have already shown up. This can be a great way to get them thinking about the lack of independence in game play.

On slide 6 is another animation of the game that can then be used to reinforce game play for students.

**Lesson Plan:**

Sticky Note #1

Now that students understand the game, ask them to use their intuition to predict the odds of winning this four-card Frustration game.

Draw a segmented interval on the whiteboard, similar to the one below, or use the interval provided on slide 8.

Be sure to leave room for at least one more of these intervals for another sticky-note vote later. You may want to have three total intervals, although the third sticky-note vote described later could be facilitated as a discussion without the actual placement of sticky notes.

Then ask students to write a verbal (qualitative) and numeric estimate on a sticky note (e.g., “unlikely, 15%”) and place it on the interval. You may want to model this for students to be sure that they place it on the appropriate location based on their percentage rather than just under the displayed words.

Have students come to the board to place their predictions on the interval, then discuss the range of predictions that is displayed.

Main Activity:

Have students work in pairs. Distribute cards (enough for at least a “deck” of four unique ranks for each pair), and one handout per person. (A deck of playing cards is recommended, but the alternative is to use the digital version available on Desmos^{®}^{.}) Although student pairs will have the same data, each individual having space to record the data themselves and think about what's going on can be helpful.

Student Handout

Student pairs will play the game 10 times and record their results. Using the slides, discuss the difference between experimental (or empirical) and theoretical probability and that the focus of this lesson will be looking at experimental probability.

Note that once students lose the game, they do not have to continue drawing cards and filling out the rest of that row. They can instead just write lose or x or however they are marking losses in the last column.

When students finish playing 10 rounds, they should fill out the bottom right corner of the table (total) with the number of wins written as a fraction and percentage of the number of games they played.

Have students share their results with at least one other pair nearby and discuss whether their results matched their predictions or not, and why they think that was the case.

__Sticky Note #2__

On the basis of their experimental results and their discussions with their peers, students should *individually* make new sticky notes with updated predictions on the likelihood of winning four-card Frustration. Draw or display a new interval on the board near the first one and have students come to the board to place their second sticky note.

After students have placed their new predictions on the second interval, have them discuss in pairs or small groups what they notice about the new predictions compared with the previous ones. For example, are they more grouped together? Did they shift higher or lower? Ideally, we hope that the sticky notes will have less of a spread and that they are grouped closer to the theoretical probability. As well as discussing observations, also discuss *why* students think the changes that they observed occurred. Emphasize how they were able to learn more about the likelihood of winning through playing.

Now that students have made two predictions and played the game, discuss how we might make even more accurate predictions. Students may suggest things that lean toward theoretical probability such as throwing out calculations for probability that may or may not be accurate. For the conversation, encourage students to make suggestions without validating if any described calculations are correct/incorrect. Instead, focus on reiterating if their idea would help find the *experimental* or *theoretical* probability.

__Experimental Probability Reveal:__

After students have shared several ideas, emphasized these two key strategies: (1) pooling the results of the classroom to look at the results from playing more games than just 10, and (2) just playing more games. Both ideas hit on the law of large numbers. Although you do not need to explicitly state the law of large numbers at this point, reiterate the idea that playing more will help us make more accurate predictions.

Finally, tell students that you, in fact, did play more games. You played one million games! Use slide 19 to show them a graph of the results from playing one million games and help students to interpret the 37.45 percent experimental probability of you (or, really, the computer) winning about 375,000 times out of one million games.

Have a discussion about how accurate they think 37.45 percent is as a prediction for winning.

Wrap-Up:

To give students a chance to apply these ideas one more time, ask them how likely they think it would be to win this game if they played with all 13 unique ranks. This is an opportunity to make one more sticky-note graph of predictions, or use a show of hands (e.g., “Raise your hand if you think it's between 0 and 20 percent, 20 and 40 percent,” etc.) and discuss where the majority of predictions are (center) and the range of predictions that were made (spread).

Using the Desmos activity or 10 physical cards (Desmos tends to be faster because there is no need to physically shuffle), as a class, play the 13-card game 10 times and have students (or one student tallying on the board) keep track of wins and total games played. Encourage students to count along as cards are dealt (“Ace, 2, 3, . . .”) to build excitement for wins/losses.

After playing 10 times calculate the percentage of wins. Quickly discuss or ask for a show of hands if students would change their original prediction, and, if you have time, invite a few students to share why or why not.

Tell students that you, again, played 1one million times last night and use the slides to show students the results of playing one million times, which resulted in an experimental probability of 36.73 percent.

Conclude by explicitly stating that the law of large numbers tells us that as the number of repeated trials (in this case, number of games played) gets very large, we can expect the winning percentage to be close to the “real” (theoretical) probability of winning the game. Explain that this is what the students were doing: Each time they played more games, they were using the idea of the law of large numbers to hopefully get a better estimate of the theoretical probability.

Finally, you can use the slides to share how likely it is to win the game if we play with all 52 cards in the deck, which is *much* harder to win, with a probability of less than 2 percent.

To end the class, distribute the exit ticket to each student. You may consider printing a copy for each student or sharing the Google Doc^{®} and forcing a copy for each student to submit virtually.

Exit Ticket