Lesson 4

Estimating Probabilities Through Repeated Experiments

4.1: Decimals on the Number Line (5 minutes)

Warm-up

The purpose of this warm-up is for students to practice placing numbers represented with decimals on a number line and thinking about probabilities of events that involve the values of the numbers. In the following activity, students are asked to graph points involving probabilities that are represented by numbers similar to the ones in this activity.

Launch

Arrange students in groups of 2. Give students 2 minutes of quiet work time followed by time to share their responses with a partner. Follow with a whole-class discussion. 

Student Facing

  1. Locate and label these numbers on the number line.

    1. 0.5
    2. 0.75
    3. 0.33
    4. 0.67
    5. 0.25

    Number line with tick marks at 0 and 1
  2. Choose one of the numbers from the previous question. Describe a game in which that number represents your probability of winning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select some partners to share their responses and methods for positioning the points on the number line. If time allows, select students to share a chance event for each of the values listed.

4.2: In the Long Run (20 minutes)

Activity

This activity begins to answer the question brought up in the previous lesson about finding the probability when the sample space is not available. Students have the opportunity to use this experiment for which the sample space is available to check its agreement and estimate based on repeating the experiment many times (MP8).

Students make the connection between probability and the fraction of outcomes for which the event occurs in the long-run. This activity highlights that a probability describes what happens in the the long run and that it does not guarantee that the event will occur a specific number of times after any specific number of trials. For example, an event that has probability 0.6 means that the event will occur about 60% of the time in the long run, but it does not mean that it will occur exactly 60 times when the experiment is performed 100 times.

Launch

Arrange students in groups of 3. Provide one standard number cube for each group. Following the teacher demonstration, allow 10 minutes for group work, followed by a whole-class discussion. 

Demonstrate how to compute and plot the current fraction of the times an event occurs.

Classes using the digital version have an applet available that automates the computation and the graphing, allowing students to focus on the probabilities.

Display the table and graph for all to see as an example of how to fill in the table and graph the results.

roll number rolled total number of wins for Mai fraction of games that are wins
1 5 0 0
2 1 1 \(\frac{1}{2} = \) 0.50
3 2 2 \(\frac{2}{3} \approx\) 0.67
4 4 2 \(\frac{2}{4} =\) 0.50
Graph. Number of rolls. Fraction of games played that are wins. 

To help students understand the graph, consider asking these questions. Ask students why the \(y\)-axis only shows 0 to 1. Ask students what the point at \((3, 0.66)\) represents.

Student Facing

Mai plays a game in which she only wins if she rolls a 1 or a 2 with a standard number cube.

  1. List the outcomes in the sample space for rolling the number cube.

  2. What is the probability Mai will win the game? Explain your reasoning.

  3. If Mai is given the option to flip a coin and win if it comes up heads, is that a better option for her to win?

  4. Begin by dragging the gray bar below the toolbar down the screen until you see the table in the top window and the graph in the bottom window. This applet displays a random number from 1 to 6, like a number cube. Mai won with the numbers 1 and 2, but you can choose any two numbers from 1 to 6. Record them in the boxes in the center of the applet.

    Click the Roll button for 10 rolls and answer the questions.

     
  5. What appears to be happening with the points on the graph?

    1. After 10 rolls, what fraction of the total rolls were a win?
    2. How close is this fraction to the probability that Mai will win?
  6. Roll the number cube 10 more times.  Record your results in the table and on the graph from earlier.

    1. After 20 rolls, what fraction of the total rolls were a win?
    2. How close is this fraction to the probability that Mai will win?

Student Response

For access, consult one of our IM Certified Partners.

Launch

Arrange students in groups of 3. Provide one standard number cube for each group. Following the teacher demonstration, allow 10 minutes for group work, followed by a whole-class discussion. 

Demonstrate how to compute and plot the current fraction of the times an event occurs.

Classes using the digital version have an applet available that automates the computation and the graphing, allowing students to focus on the probabilities.

Display the table and graph for all to see as an example of how to fill in the table and graph the results.

roll number rolled total number of wins for Mai fraction of games that are wins
1 5 0 0
2 1 1 \(\frac{1}{2} = \) 0.50
3 2 2 \(\frac{2}{3} \approx\) 0.67
4 4 2 \(\frac{2}{4} =\) 0.50
Graph. Number of rolls. Fraction of games played that are wins. 

To help students understand the graph, consider asking these questions. Ask students why the \(y\)-axis only shows 0 to 1. Ask students what the point at \((3, 0.66)\) represents.

Student Facing

Mai plays a game in which she only wins if she rolls a 1 or a 2 with a standard number cube.

  1. List the outcomes in the sample space for rolling the number cube.
  2. What is the probability Mai will win the game? Explain your reasoning.
  3. If Mai is given the option to flip a coin and win if it comes up heads, is that a better option for her to win?
  4. With your group, follow these instructions 10 times to create the graph.

    • One person rolls the number cube. Everyone records the outcome.

    • Calculate the fraction of rolls that are a win for Mai so far. Approximate the fraction with a decimal value rounded to the hundredths place. Record both the fraction and the decimal in the last column of the table.

    • On the graph, plot the number of rolls and the fraction that were wins.

    • Pass the number cube to the next person in the group.

    roll outcome total number of
    wins for Mai
    fraction of games
    played that are wins
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    A blank coordinate grid with the origin labeled “O.” 
  5. What appears to be happening with the points on the graph?
    1. After 10 rolls, what fraction of the total rolls were a win?
    2. How close is this fraction to the probability that Mai will win?
  6. Roll the number cube 10 more times. Record your results in this table and on the graph from earlier.
    roll outcome total number of
    wins for Mai
    fraction of games
    played that are wins
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    1. After 20 rolls, what fraction of the total rolls were a win?
    2. How close is this fraction to the probability that Mai will win?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may not notice a pattern in the graph. Ask if they can see a pattern with the decimal values for the fraction of wins in their table. If their data does not fit the expected pattern, tell them that this is not typical and ask them to look at another group's results.

Activity Synthesis

The purpose of this discussion is for students to understand that computing the fraction of the time an event occurs can be used to estimate the probability of the event and that more repetitions should make the estimation more accurate. 

Select some students to share their answer and reasoning for the second question. If it is not mentioned by students, tell them that when there is more than one outcome that is in the desired event, then the probability of that event is the number of outcomes in the desired event divided by the number of outcomes in the sample space. In this example, there are 2 outcomes that win (a roll of 1 or 2) and 6 outcomes in the sample space, so the probability of winning is \(\frac{2}{6}\) which is equivalent to \(\frac{1}{3}\).

Collect the number of 1s and 2s for each group and compute the fraction for the whole class with all the data. The value should be very close to \(\frac{1}{3}\).

Select students to share their thoughts on what appears to be happening with the points on their graph. (They are leveling out at 0.33.) If students struggle with noticing that the points are leveling out at a \(y\) value of 0.33, ask them to draw a horizontal line on their graphs at their answer for the probability they got in the second question. 

Ask the class how many times the entire class rolled number cubes. Then ask, "Based on the probability predicted in the second question, how many times do we expect the class to have simulated a win for Mai? How does this compare to the actual number of wins the class rolled."

A probability tells you how likely an event is to occur. While it is not guaranteed to be an exact match, if the chance experiment is repeated many times, we expect the fraction of times that an event occurs to be fairly close to the calculated probability.

Representation: Internalize Comprehension. Activate or supply background knowledge about finding patterns with decimal values for the fraction of wins in the last statement. Some students may benefit from a demonstration of how to approximate fractions with decimal values to graph. Invite students to engage in the process by offering suggested directions as you demonstrate.
Supports accessibility for: Visual-spatial processing; Organization
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate and/or revoice 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 will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

4.3: Due For a Win (10 minutes)

Activity

This activity gives students the opportunity to see that an estimate of the probability for an event should be close to what is expected from the exact probability in the long-run; however, the outcome for a chance event is not guaranteed and estimates of the probability for an event using short-term results will not usually match the actual probability exactly (MP6).

Launch

Tell students that the probability of a coin landing heads up after a flip is \(\frac{1}{2}\)

Give students 5 minutes of quiet work time followed by a whole-class discussion.

Student Facing

  1. For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning.

    1. You flip the coin once, and it lands heads up.
    2. You flip the coin twice, and it lands heads up both times.
    3. You flip the coin 100 times, and it lands heads up all 100 times.
  2. If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning.
  3. If you flip the coin 100 times, what are some other results that would not be surprising?
  4. You’ve flipped the coin 3 times, and it has come up heads once. The cumulative fraction of heads is currently \(\frac{1}{3}\). If you flip the coin one more time, will it land heads up to make the cumulative fraction \(\frac{2}{4}\)?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is for students to recognize that the actual results from repeating an experiment should be close to the expected probability, but may not match exactly.

For the first problem, ask students to indicate whether or not they think each result seems surprising. For the second and third questions, select several students to provide answers and display for all to see, then create a range of values that might not be surprising based on student responses. Ask the class if they agree with this range or to provide a reason the range is too large. It is not important for the class to get exact values, but a general agreement should arise that some range of values makes sense so that there does not need to be exactly 50 heads from the 100 flips.

An interesting problem in statistics is trying to define when things get "surprising." Flipping a fair coin 100 times and getting 55 heads should not be surprising, but getting either 5 or 95 heads probably is. (Although there is not a definite answer for this, a deeper study of statistics using additional concepts in high school or college can provide more information to help choose a good range of values.)

Explain that a probability represents the expected likelihood of an event occurring for a single trial of an experiment. Regardless of what has come before, each coin flip should still be equally likely to lands heads up as tails up.

As another example: A basketball player who tends to make 75% of his free throw shots will probably make about \(\frac{3}{4}\) of the free throws he attempts, but there is no guarantee he will make any individual shot even if he has missed a few in a row.

Conversing, Reading: MLR2 Collect and Display. As students share whether each result is surprising or not, write down the words and phrases students use to explain their reasoning. Listen for students who state that the actual results from repeating an experiment should be close to the expected probability. As students review the language collected in the visual display, encourage students to revise and improve how ideas are communicated. For example, a phrase such as: “If you flip a coin 100 times, it is impossible for the coin to land heads up all 100 times” can be improved with the phrase “If you flip a coin 100 times, it is very unlikely for the coin to land heads up all 100 times, but it is possible.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Consider asking these questions:

  • "You conduct a chance experiment many times and record the outcomes. How are these outcomes related to the probability of a certain event occurring?" (The fraction of times the event occurs after many repetitions should be fairly close to the expected probability of the event.)
  • "What is the probability of rolling a 2, 3, or 4 on a standard number cube? If you roll 3 times and none of them result in a 2, 3, or 4, does the probability of getting one of those values change with the next roll?" (The probability is 0.5 since 3 outcomes out of 6 possible are in the event. The probability should not change after 3 times. If a 2, 3, or 4 does not appear after a lot of rolls—say, 100—then we might suspect the number cube of being non-standard.)
  • "The probability of getting the flu during flu season is \(\frac{1}{8}\). If a family has 8 people living in the same house, is it guaranteed that one of them will get the flu? If a country has 8 million people, about how many do you expect will get the flu? Does this number have to be exact?" (No, it is very possible that none of the people in the family will get the flu and also possible that more than 1 person will get the flu. We might expect about 1 million people in the country to get the flu, but this is probably not exact.)

4.4: Cool-down - Fiction or Non-fiction? (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

A probabilityfor an event represents the proportion of the time we expect that event to occur in the long run. For example, the probability of a coin landing heads up after a flip is \(\frac12\), which means that if we flip a coin many times, we expect that it will land heads up about half of the time.

Even though the probability tells us what we should expect if we flip a coin many times, that doesn't mean we are more likely to get heads if we just got three tails in a row. The chances of getting heads are the same every time we flip the coin, no matter what the outcome was for past flips.