# Lesson 7

Related Events

## 7.1: Drawing Crayons (5 minutes)

### Warm-up

The mathematical purpose of this activity is to show students that some events can depend on one another. It is not important that students get a single, correct solution for the second problem.

### Student Facing

A bag contains 1 crayon of each color: red, orange, yellow, green, blue, pink, maroon, and purple.

1. A person chooses a crayon at random out of the bag, uses it for a bit, then puts it back in the bag. A second person comes to get a crayon chosen at random out of the bag. What is the probability the second person gets the yellow crayon?
2. A person chooses a crayon at random out of the bag and walks off to use it. A second person comes to get a crayon chosen at random out of the bag. What is the probability the second person gets the yellow crayon?

### Activity Synthesis

The goal of this discussion is to introduce the terms independent events and dependent events in the context of the crayon problem. Ask students “How are the questions different?” (In the first question it did not matter what crayon was used first, but in the second question it did matter.) Tell students that independent events are two events from the same experiment for which the probability of one event is not affected by whether the other event occurs or not. Dependent events are two events from the same experiment for which the probability of one event relies on the result of the other event.

Here are some questions for discussion.

• “Which of the two events is independent?” (Choosing the first crayon, then replacing it and choosing the second crayon.)
• “Which of the two events is dependent?” (Choosing the first crayon, then without replacing it choosing the second crayon.)
• “When the crayon was not replaced, the two events are dependent. Describe what it means for the events to be dependent using the crayon context.” (It means that you can’t figure out the probability of what color crayon will be second without knowing what color crayon was chosen first.)
• “When the crayon was replaced, the two events are independent. Describe what it means for the events to be independent using the crayon context.” (It means that you can find the probability of either event separately. For example, in the first question when you pick a crayon from the bag the second time it does not matter what was picked the first time because you know the color of all six crayons that are in the bag.)

## 7.2: Choosing Doors (30 minutes)

### Activity

In this lesson, students explore variations of the classic "Monty Hall" problem to understand what it means for events to be dependent or independent. This exploration introduces a context that is not intuitive to many people prior to a deeper study of conditional probability. In these examples, one event (winning the prize) depends on the outcome of another event (choosing to stay or switch from the chosen door).

### Launch

Arrange students in groups of 2. Remind students that it is important to play the games fairly. The host should not try to help their partner nor should they try to trick them. As with most data collection, it is important to not try to influence the results.

Demonstrate the games by playing the role of host while a student is the contestant. Allow students to play the first game for 10 minutes before telling students to move on to the questions and second game.

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

1. On a game show, a contestant is presented with 3 doors. One of the doors hides a prize and the other two doors have nothing behind them.
• The contestant chooses one of the doors by number.
• The host, knowing where the prize is, reveals one of the empty doors that the contestant did not choose.
• The host then offers the contestant a chance to stay with the door they originally chose or to switch to the remaining door.
• The final chosen door is opened to reveal whether the contestant has won the prize.

Choose one partner to play the role of the host and the other to be the contestant. The host should think of a number: 1, 2, or 3 to represent the prize door. Play the game keeping track of whether the contestant stayed with their original door or switched and whether the contestant won or lost.

Switch roles so that the other person is the host and play again. Continue playing the game until the teacher tells you to stop. Use the table to record your results.

stay switch total
win
lose
total
1. Based on your table, if a contestant decides they will choose to stay with their original choice, what is the probability they will win the game?
2. Based on your table, if a contestant decides they will choose to switch their choice, what is the probability they will win the game?
3. Are the two probabilities the same?

2. In another version of the game, the host forgets which door hides the prize. The game is played in a similar way, but sometimes the host reveals the prize and the game immediately ends with the player losing, since it does not matter whether the contestant stays or switches.

Choose one partner to play the role of the host and the other to be the contestant. The contestant should choose a number: 1, 2, or 3. The host should choose one of the other two numbers. The contestant can choose to stay with their original number or switch to the last number.

After following these steps, roll the number cube to see which door contains the prize:

• Rolling 1 or 4 means the prize was behind door 1.
• Rolling 2 or 5 means the prize was behind door 2.
• Rolling 3 or 6 means the prize was behind door 3.

Play the game keeping track of whether the contestant stayed with their original door or switched and whether the contestant won or lost.

Switch roles so that the other person is the host and play again. Continue playing the game until the teacher tells you to stop. Use the table to record your results.

stay switch total
win
lose
total
1. Based on your table, if a contestant decides they will choose to stay with their original choice, what is the probability they will win the game?
2. Based on your table, if a contestant decides they will choose to switch with their original choice, what is the probability they will win the game?
3. Are the two probabilities the same?

### Student Facing

#### Are you ready for more?

In another version of the game, the contestant is presented with 5 doors. One of the doors hides a prize and the other four doors have nothing behind them.

• The contestant chooses 3 doors by number.
• The host, knowing where the prize is, reveals 3 of the doors that have nothing behind them. Two of the doors that the contestant has chosen that are empty and one of the other doors that are empty.
• The host then offers the contestant a chance to stay with the door they originally chose or to switch to the remaining door.
• The final chosen door is opened to reveal whether the contestant has won the prize.

Choose one partner to play the role of the host and the other to be the contestant. The host should think of a number: 1, 2, 3, 4, or 5 to represent the prize door. Play the game keeping track of whether the contestant stayed with their original door or switched and whether the contestant won or lost.

Switch roles so that the other person is the host and play again. Continue playing the game until the teacher tells you to stop. Use the table to record your results.

stay switch total
win
lose
total
1. Based on your table, if a contestant decides they will choose to stay with their original choice, what is the probability they will win the game?
2. Based on your table, if a contestant decides they will choose to switch with their original choice, what is the probability they will win the game?
3. Are the two probabilities the same?

### Anticipated Misconceptions

Some students may not understand why the probability of winning when choosing to switch doors is $$\frac{2}{3}$$. Prompt students to look at the three possibilities of how the winning (W) and losing doors (L) can be arranged:

• WLL, LWL, or LLW

Emphasize that choosing to stay means that the probability of winning is equal to the probability that your first choice is the winning door. The probability that your first choice is the winning door is $$\frac{1}{3}$$. This also means that the probability that your first choice is a losing door is $$\frac{2}{3}$$.  If you start with a losing door that means that the remaining two doors are L and W. If you choose to switch, the host eliminates the losing door, and only the winning door remains. This means that if you start on a losing door and switch then you are guaranteed to win. Therefore, the probability of winning when choosing to switch is $$\frac{2}{3}$$ (the probability that your first choice is a losing door).

### Activity Synthesis

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

Poll the class to collect results, and display a table that shows all of the results for each game. Tell students, “In the first game, the event of a win is dependent on the event of switching since the probability of winning is different depending on whether the contestant stays or switches. In the second game, the event of a win is independent of the event of switching since the probability of winning is the same whether the contestant stays or switches.“

Here are some questions for discussion.

• “In the first game, is choosing which door to open at the start of the game dependent on anything?” (It is not dependent on anything.)
• “In the first game, is choosing to switch dependent on anything? ” (It depends on the fact that you know that the host is going to eliminate a door that is not a winner rather than a door chosen at random.)
• “In the first game, what are the different outcomes?” There are four outcomes: that you win by staying ($$\frac{1}{3}$$ of the time), lose by staying ($$\frac{2}{3}$$ of the time), win by switching ($$\frac{2}{3}$$ of the time) or lose by switching ($$\frac{1}{3}$$ of the time).)
• “Does it help you to think about this activity by making the decision to stay or switch before you play the game?” (Yes, that really helps me because we are talking about probability and the decision to switch kept getting in the way of me thinking about the probabilities. It really helps me to look at it as two different situations.)
• “In the second game, is choosing to switch dependent on anything?” (It is not dependent on anything because the host is eliminating a door at random.)
• “What are the possible outcomes for the second game?” (There are four outcomes: that you win by staying ($$\frac{1}{3}$$ of the time), lose by staying ($$\frac{2}{3}$$ of the time), win by switching (($$\frac{1}{3}$$ of the time) or lose by switching ($$\frac{2}{3}$$ of the time).)

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 they are going to stay regardless of what happens, then they must choose the correct door right at the beginning. There is a $$\frac{1}{3}$$ chance of choosing the right door.

If a player decides they are going to switch regardless of what happens, then they must choose one of the doors that have nothing behind them in order to win. If they choose either one, the host will reveal the other one and the player can switch to the correct door. Since the goal is to initially choose a door with nothing behind it, there is a $$\frac{2}{3}$$ chance 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 makes their choice, the host reveals 98 empty rooms leaving only the one the contestant chose and one other door closed, similar to the 2 left in the first game. Now consider whether the initial door that was chosen is more likely to contain the prize or the single other door the host left untouched.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. After the class has been polled and their data has been displayed in a table, present an incorrect interpretation of the likelihood of winning the game. For example, “The probability of winning is always $$\frac13$$ because only one door out of three has a prize behind it.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify that there are three possibilities of how the winning and losing doors can be arranged. For example, the author probably thought that there was only one way to win or did not realize that the doors could be arranged in three ways, WLL, LWL, or LLW, and switching doors changes the probability of winning. This will help students better understand when events are independent or dependent.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion.

• "What does it mean for two events to be independent?" (Two events are independent when the probability of one event occurring does not change whether the other event occurs or not.)
• "What does it mean for two events to be dependent?" (Two events are dependent when the probability of one event occurring is different if the other event occurs or not.)
• “There are 10 different names written on slips of paper that are placed in a hat. Your teacher picks one name out of a hat and places it on the desk. The teacher then picks another name out of the hat. Are these events dependent or independent? Explain your reasoning.” (They are dependent because the probability of picking a name the second time is dependent on what name was picked first. For example, if my name was picked first, I would have a $$\frac{0}{9}$$ chance of it being picked second. If my name was not picked first then I would have a $$\frac{1}{9}$$ chance of it being picked second.)
• “How does the situation change if the first name is placed back in the hat?” (It changes because now the events are independent. The name picked first does not change the probability of the name being picked second because all the names will be in the bag regardless of the first name picked.)
• “In science class you may have heard about independent and dependent variables. How is this related to independent and dependent events?” (In science class, the dependent variables is the one that changes in response to the independent variable. For example the temperature at which water boils depends on the pressure, so pressure is the independent variable and the temperature is the dependent variable.)
• “Can you think of any other examples that use dependence and independence?” (One example would be your performance review at a job and how much you are paid. I would hope they are dependent, but they do not have to be.)

## Student Lesson Summary

### Student Facing

When considering probabilities for two events it is useful to know whether the events are independent or dependent. Independent events are two events from the same experiment for which the probability of one event is not affected by whether the other event occurs or not. Dependent events are two events from the same experiment for which the probability of one event is affected by whether the other event occurs or not.

For example, let's say a bag contains 3 green blocks and 2 blue blocks. You are going to take two blocks out of the bag.

Consider two experiments:

1. Take a block out, write down the color, return the block to the bag, and then choose a second block. The event, "the second block is green" is independent of the event, "the first block is blue." Since the first block is replaced, it doesn’t matter what block you picked the first time when you pick a second block.
2. Take a block out, hold on to it, then take another block out. The same two events, "the second block is green" and "the first block is blue," are dependent.

If you get a blue block on the first draw, then the bag has 3 green blocks and 1 blue block in it, so $$P(\text{green})=\frac{3}{4}$$.

If you get a green block on the first draw, then the bag has 2 green blocks and 2 blue blocks in it, so $$P(\text{green}) = \frac{1}{2}$$.

Since the probability of getting a green block on the second draw changes depending on whether the event of drawing a blue block on the first draw occurs or not, the two events are dependent.

In some cases, it is difficult to know whether events are independent without collecting some data. For example, a basketball player shoots two free throws. Does the probability of making the second shot depend on the outcome of the first shot? Some data would need to be collected about how often the player makes the second shot overall and how often the player makes the second shot after making the first so that you could compare the estimated probabilities.