The purpose of this discussion is for students to gain a deeper understanding of what it means for events to be dependent or independent.

Collect the results from the class, and combine them into a table that shows all of the results for each game. 

Tell students, “For each game, let event A be the decision to switch and event B be the winning.”

Display the question, “Are the events in the first version of the game independent or dependent? Explain your reasoning.”

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to whether the events are independent or not in the first version of the game by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  “The events are independent because you’re allowed to stay or switch no matter what door you pick.”
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Display and review these criteria: 
  • What “independent or dependent events” mean
  • What the events in this situation are
  • Whether the events are independent or dependent
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Ask students,

• “If events A and B are dependent, what does that mean in this situation?” (It means that the probability of winning the game is different if you decide to switch or stay.)
• “Are the two events independent or dependent for the first version of the game? Explain your reasoning.” (They are dependent because in games where the contestant decided to stay, they won about of the time. In games where the contestant decided to switch, they won about of the time. Because the probability of winning changes depending on whether the contestant decides to stay or switch, the events are dependent.)
• “Are the two events independent or dependent for the second version of the game? Explain your reasoning.” (They are independent because the contestants won about of the time whether they decided to switch or stay. Because the probability is the same regardless of whether the contestant stays or switches, the events are independent.)

One way to explain the dependency for the first game is to imagine the decisions being made in the other order. If a player decides to stay regardless of what happens, then the player must choose the correct door right at the beginning. There is a probability of  of choosing the right door.

If a player decides to switch regardless of what happens, then the player must choose one of the doors that has nothing behind it in order to win. If the player chooses either of the doors with nothing behind it, the host will reveal the other door with nothing behind it, and the player switches to the prize door. When deciding to switch doors, the goal is to initially choose a door with nothing behind it, so there is a probability of of winning the game.

A second way to understand the reasoning is to imagine a larger game with 100 doors to choose from. After the contestant chooses a door, the host opens all but one of the remaining doors to reveal 98 empty rooms leaving only the door the contestant chose and one other door closed, similar to the 2 left in the first game. Now consider which door is more likely to contain the prize—the initial door that was chosen or the single other door that the host left untouched. (In this version of the game, a contestant who decides to stay has a probability of winning. The contestant who decides to switch has a probability of winning.)

In the second version of the game, the host does not provide the contestant with any additional information about where the prize is. Because the host can accidentally open the prize door and immediately end the game with a loss, the probability of winning does not depend on the contestant’s choice of staying or switching.

Math Community

Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘.’ It really helped me/my group because .”
• “I picked the norm ‘.’ During the activity, I realized the norm ‘’ would be a better focus because .”