Estimating Probabilities through Repeated Experiments
Accelerated 6
14.1
Warm-up
Instructional Routines
None
Materials
None
Activity Narrative
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. Values between 0 and 1, such as the ones found here, are related to probabilities students will see in later activities. They will also need to be able to plot probabilities.
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.
Activity
None
Locate and label these numbers on the number line.
0.5
0.75
0.33
0.67
0.25
Choose one of the numbers from the previous question. Describe a game in which that number represents your probability of winning.
Student Response
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Building on Student Thinking
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.
14.2
Activity
Standards Alignment
Building On
7.NS.A.2.d
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
This activity begins to address how to find the probability when the sample space is not known. 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 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.
In the digital version of the activity, students use an applet to play and plot the cumulative fraction of wins for a game. The applet allows students to focus on visualizing the cumulative fraction as narrowing in on the probability. Use the digital version if students would benefit from concentrating on the current mathematical goals rather than reviewing how to plot values on a coordinate plane.
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 probability that Mai will win the game. If it is not mentioned by students, tell them that the probability of an event is the number of outcomes in the given 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 , which is equivalent to .
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 .
Select students to share their thoughts on what appears to be happening with the points on their graph. (They are leveling out around 0.33.) If students struggle with noticing that the points are leveling out at a -value around 0.33, ask them to draw a horizontal line on their graph 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?”
Collect the number of wins for each group and compute the fraction for the whole class with all the data. The value should be very close to .
A probability tells us 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.
MLR8 Discussion Supports. Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “at the end it goes like this” as “in the long run the graph levels out.” Advances: Speaking, Representing
Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use the table or the digital applet, some students may benefit from a demonstration or access to step-by-step instructions. Supports accessibility for: Organization, Memory, Attention
14.3
Activity
Standards Alignment
Building On
Addressing
7.RP.A
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
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 .
Give students 5 minutes of quiet work time, and follow with a whole-class discussion.
Activity
None
For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning.
You flip the coin once, and it lands heads up.
You flip the coin twice, and it lands heads up both times.
You flip the coin 100 times, and it lands heads up all 100 times.
If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning.
If you flip the coin 100 times, what are some other results that would not be surprising?
You’ve flipped the coin 3 times, and it has landed heads up once. The cumulative fraction of heads is currently . If you flip the coin one more time, will it land heads up to make the cumulative fraction ?
Student Response
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Building on Student Thinking
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 a later course 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 land heads up as tails up.
As another example: A basketball player who tends to make 75% of their free throw shots will probably make about of the free throws they attempt, but there is no guarantee they will make any individual shot even if they have missed a few in a row.
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 . 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 possible that none of the people in the family will get the flu, and it is 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.)
Student Lesson Summary
A probability for 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 , 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 outcomes were for past flips.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Arrange students in groups of 3. Provide 1 standard number cube for each group. Following the teacher demonstration, allow 10 minutes for group work, followed by a whole-class discussion.
Read or ask a student to read the problem stem explaining the win condition for Mai.
Display the table and graph for all to see as an example of how to fill in the table and graph the results. Demonstrate how to compute and plot the current fraction of the games that are wins. Display possible results such as these.
roll
outcome
total number of
wins for Mai
fraction of games
played that are wins
1
5
0
0
2
1
1
0.50
3
2
2
0.67
4
4
2
0.50
To help students understand the graph, consider asking these questions.
“Why does the -axis only show 0 to 1?” (It represents the cumulative fraction of games that are won. It only makes sense to win all of the time with 1, none of the time with 0, or with values between.)
“What does the point at represent?” (It means that after 3 times playing the game, Mai has won about 67% of the time.)
Activity
None
Mai plays a game in which she only wins if she rolls a 1 or a 2 with a standard number cube.
List the outcomes in the sample space for rolling the number cube.
What is the probability Mai will win the game? Explain your reasoning.
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?
With your group, follow these instructions 10 times to create the graph.
One person rolls the number cube. Everyone records the outcome in the table.
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 of games that are wins.
Pass the number cube to the next person in the group so they can repeat these steps.
Describe how the graph changes as the number of rolls increases.
After 10 rolls, what fraction of the total rolls are a win?
How close is this fraction to the probability that Mai will win?
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.” The horizontal axis is labeled “number of rolls” and the numbers 1 through 20 are indicated. The vertical axis is labeled “fraction of games played that are wins” and the numbers 0 through 1, in increments of zero point 1, are indicated.
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
After 20 rolls, what fraction of the total rolls are wins?
How close is this fraction to the probability that Mai will win?
Student Response
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Building on Student Thinking
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.
