Week 6 Data Lab: What Is the Larger Die Worth?

Part 1 — The Big Picture

This week you learned that a number set by chance has a probability distribution, an expected value that the long run settles toward, and a standard deviation that says how far one result typically swings. In this lab you build a new random variable from two dice, the larger of the two faces, work out its distribution, expected value, and standard deviation from the 36 equally likely outcomes, and then check the model against 500 rounds you roll in a spreadsheet. A chatbot then tries the same job, and you catch where it goes wrong.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: What is the larger of two dice worth on average over many rounds, how far does one round typically swing from that average, and does a 500-round simulation agree with the model?

Before you begin — hypothesis: Before you build anything, write one sentence in this form: If I roll two dice 500 times and keep the larger face each time, then the average larger face will be about ___, because ___.

Workshop data: Larger-of-two-dice rounds

Source: Rounds you generate yourself in a spreadsheet: each round rolls two fair dice and keeps the larger face, alongside the 36-outcome model you build by hand.

• Rows: 500 simulated rounds, one per spreadsheet row; the first scaffold table summarizes the model in 7 rows, six larger faces plus a total row
• Columns: first die, second die, and the larger face
• Model: 36 equally likely ordered outcomes of two fair dice

Part 3 — Materials & Procedure

Tools and materials

• Google Sheets (free) or Excel
• A document or notebook for your written answers

Ways of working

• interpret-real-data
• compute-and-plot
• structured-observation

Procedure

1. Build the 36-outcome model. In a blank sheet, type 1 to 6 across cells B1 to G1 for the second die and down cells A2 to A7 for the first die. In B2 type =MAX($A2,B$1), then fill it right to G2 and down to row 7. Tool: Google Sheets (free) or Excel. Check: A 6-by-6 grid of larger faces appears, with only whole numbers from 1 to 6.
2. Count each larger face. In I2 to I7 type the faces 1 to 6. In J2 type =COUNTIF($B$2:$G$7,I2) and fill down to J7. In K2 type =J2/36 and fill down to K7. Copy the counts and probabilities into the first scaffold table. Tool: Google Sheets (free) or Excel. Check: The six counts add to 36, and =SUM(K2:K7) shows 1.
3. Compute the model's expected value and spread. In an empty cell type =SUMPRODUCT(I2:I7,K2:K7) for the expected value. In the next cell type =SUMPRODUCT((I2:I7-M2)^2,K2:K7), where M2 is the cell holding the expected value, for the variance, and then =SQRT of that cell for the standard deviation. Tool: Google Sheets (free) or Excel. Check: The expected value lies between 1 and 6, and the standard deviation is a positive number smaller than 5.
4. Simulate 500 rounds on a new sheet. In A2 type =RANDBETWEEN(1,6), in B2 type =RANDBETWEEN(1,6), and in C2 type =MAX(A2,B2). Select A2 to C2 and fill down to row 501. Tool: Google Sheets (free) or Excel. Check: Rows 2 to 501 hold 500 rounds, and every value in column C is a whole number from 1 to 6.
5. Summarize your rounds. Type =AVERAGE(C2:C501) and =STDEV.S(C2:C501), and for each face from 1 to 6 type =COUNTIF(C2:C501,face)/500 to get its relative frequency. Copy or paste these as values so they stop changing, then record them in the second scaffold table. Tool: Google Sheets (free) or Excel. Check: The six relative frequencies add to 1.
6. Make a comparison chart. Put the six model probabilities and the six simulated relative frequencies in two side-by-side columns, select them, and use Insert, then Chart, choosing a column chart with one pair of bars for each larger face. Give the chart a title and label both axes in words. Tool: Google Sheets (free) or Excel. Check: Each larger face shows two bars, one from the model and one from your rounds, and the axis labels say what each bar measures.
7. Answer the analysis questions in your document, then paste your two tables into an approved chatbot with the prompt in Part 6 and check its answer against the scripted checks. Tool: A document or notebook for your written answers. Check: You have five written answers and a list of the chatbot's errors with corrections.

Fallback route

When to use it: Use this route if no spreadsheet is available or the formulas will not run.

Build the 36-outcome grid by hand on paper, and use the model data below, which reports the counts from one completed run of 500 rounds, in place of your own simulation. Compute the run's average and standard deviation from the counts, complete both scaffold tables, every analysis question, and the critique step, and submit the same items for equal credit.

This route earns the same credit as the primary route.

These counts come from one completed 500-round run, so you can finish the comparison without a spreadsheet.

Fallback model data

Larger face	Rounds with that larger face (out of 500)
1	16
2	38
3	57
4	105
5	129
6	155
Total	500

Part 4 — The Scaffold

Build the model from the 36 equally likely outcomes first, because every later comparison checks your rounds against it.

Student scaffold table 1

Larger face	Outcomes out of 36	Probability	Face times probability
1
2
3
4
5
6
Total

Set the model beside your 500 rounds, because a simulation only means something when you compare it with the model.

Student scaffold table 2

Quantity	From the 36-outcome model	From your 500 rounds
Expected value or average
Standard deviation
Chance or share of a larger face of 6

Part 5 — Analysis Questions

1. Use your first table to show that the distribution of the larger face is a legitimate probability distribution, then report its expected value and standard deviation from your formulas, rounded to two decimals, with the weighted sum written out.
2. Compare your simulated average, standard deviation, and share of 6s with the model values. How close is each one, and in which direction does each miss?
3. Explain why the expected larger face is not a value that one round can show, and say in one sentence what it does describe.
4. Describe the shape of the larger-face distribution using the shape words from Week 2, and compare it with the flat shape of one die and the symmetric shape of the two-dice sum. Where does the expected value sit relative to the pile of probability?
5. Choose one and answer it in 3–4 sentences. (a) Your rounds came from a spreadsheet's random-number function, not real dice: name one way they could differ from real rolls, and describe how you would check. (b) 500 rounds is one short run: say what you would expect to change, and what to stay about the same, if you ran 50 rounds or 5,000 rounds instead.

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I've pasted my completed Week 6 Data Lab tables above. Two fair dice are rolled and only the larger of the two faces is kept. Please do five things. First, give the probability distribution of the larger face, with the probability of each value from 1 to 6. Second, give its expected value and standard deviation. Third, describe the shape of the distribution. Fourth, as a look back at Week 4, say whether the correlation of r = 0.8712 between flipper length and body mass in the Palmer penguins field log shows that longer flippers cause heavier penguins. Fifth, as a stretch question for later in the course, say how sure we can be about the true long-run average of the larger face, based on my 500 rounds.

Test the response for these likely failures:

• Chatbots often give the chance that the larger face is 6 as 1 ⁄ 36, counting only the outcome (6, 6), or as 1 ⁄ 6, forgetting the number of ways a larger face of 6 can happen.
• Chatbots often report a simulated average or standard deviation as if they had run your 500 rounds, or reuse one die's expected value of 3.5 for the larger face.
• Asked about shape, chatbots often call the distribution right-skewed because most rounds pile up at the high faces.
• Asked about the penguins, chatbots often say longer flippers make penguins heavier because r = 0.8712 is a strong correlation.
• Asked how sure we can be, chatbots often compute an interval from your run and say there is a 95% chance the true average is inside it.

In 2–3 sentences, say what the chatbot got right and correct at least two of its errors, citing evidence from your own tables.

Equivalent route (no AI required)

No AI account is required for this lab. Instead of the chatbot, critique this flawed analysis, printed here in full: "The larger of two dice is 6 only when both dice show 6, so the chance of a larger face of 6 is 1 ⁄ 36. The expected larger face is 3.5, the same as one die, and my simulated average came out at exactly 4.5000 with a standard deviation of exactly 1.0000. The distribution is right-skewed, because most rounds land on the high faces. The penguin flipper lengths and body masses have r = 0.8712, so longer flippers make penguins heavier. There is a 95% chance that the true average larger face is between 4.4 and 4.5." Mark it up: quote each flawed sentence, name what is wrong (a dropped number of ways, an invented statistic, a flipped skew direction, cause read from association, or a misread confidence statement), cite the evidence from your tables, and write a corrected sentence beneath it; for the last sentence, say what the course will check later. This route assesses the same objectives (Objective 4 and Objective 2) as the chatbot critique, takes comparable effort (about 30 minutes and 200–300 words), is scored by the same scoring criteria for the same credit, and its own submission artifact is the marked-up flawed analysis, posted in place of the chatbot share link and report.

Part 7 — What to Submit

• Both completed scaffold tables
• The comparison chart, or a labelled sketch of it
• Written answers to all five analysis questions, including your hypothesis sentence
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Data Lab report on the larger of two dice: the 36-outcome model table with its expected value and standard deviation, a 500-round simulation summarized beside the model with a comparison chart, written answers to five analysis questions, and a critique of a chatbot's analysis of the same random variable, or, on the equivalent route, a marked-up correction of the printed flawed analysis. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Building and checking the 36-outcome model	The grid is built, or drawn by hand on the fallback route; all six counts and probabilities are recorded; and both legitimacy checks are shown, with counts adding to 36 and probabilities adding to 1.	The model is built, but one or two counts are wrong, or one legitimacy check is missing.	No model table is recorded, or its counts are invented rather than counted.
Expected value and standard deviation by formula	The expected value and the standard deviation of the larger face are computed with the weighted sum written out, rounded at the end, and stated in the face's units.	One of the two is computed correctly, or both are attempted with an error such as dividing by a count instead of weighting by probabilities.	Neither value is computed, or the values are copied without any method.
Simulation and comparison	500 rounds are simulated, or the fallback counts are used; the average, standard deviation, and share of 6s are recorded beside the model values; and the chart or labelled sketch compares model and simulation face by face.	The simulation is summarized, but one quantity, the comparison, or the chart is missing or mislabelled.	No simulation summary or comparison is submitted.
Interpretation of center, spread, and shape	Analysis questions 2 to 4 are answered accurately: the size and direction of each simulation miss, the expected value read as a long-run average rather than a single round, and the left skew named for its tail and compared with the other two shapes.	The questions are attempted, but one answer reads the expected value as a forecast, names the skew by its pile, or leaves out a comparison.	The interpretation questions are missing or do not address the larger face.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors are quoted and corrected with evidence from the student's own tables, and one correct point is acknowledged; both routes earn this band equally.	On either route, errors are named but corrected without evidence, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.
Reflection on what one simulated run can show	The chosen reflection option is answered in 3–4 sentences that connect the simulation's length or its random-number source to a specific limit on its conclusions.	The reflection is answered but stays general or does not name a specific limit.	No reflection is submitted.

Quality Gate

• Every count, probability, average, and standard deviation comes from your own formulas, your hand-built grid, or the fallback model data, never from memory or a chatbot.
• The model's counts add to 36 and its probabilities add to 1; your simulated relative frequencies add to 1.
• The simulated values are pasted as values, so they match the chart and the tables you submit.
• No statistic, count, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence from your tables.