Week 5 Data Lab: Chance in the Long Run

Part 1 — The Big Picture

This week you learned that a probability is a long-run share: one roll of a die is unpredictable, but many rolls settle into a pattern. In this lab you make that happen yourself. You roll a fair die 600 times and two dice 360 times in a spreadsheet, watch the share of sixes swing early and settle later, and check the addition rule against your own counts. Then you open the real penguin field log, compute conditional probabilities by island, and decide whether species and island are independent. Last, you check a chatbot's answers about the same chances.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: How close do simulated shares come to the chances you count from the 36 dice outcomes as the rolls pile up, and does knowing a penguin's island change the chance that it is an Adelie?

Workshop data: palmer-station-penguins

Get the data: palmer-station-penguins landing page or the direct data file — free, no login required.

Source: Measurements of 344 penguins of three species on three islands of Antarctica's Palmer Archipelago, recorded by field researchers and republished in the palmerpenguins package.

• Rows: 344 penguins
• Columns: species, island, bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g, sex, year
• Missing values: Numeric measurement columns and sex carry missing values for a small number of penguins

Part 3 — Materials & Procedure

Tools and materials

• Google Sheets (free) or Excel
• The palmerpenguins CSV file
• A document or notebook for your written answers

Ways of working

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

Procedure

1. Open a blank sheet and name the first tab Die. In cell A1 type Roll result. In A2 type =RANDBETWEEN(1,6), then copy it down to A601, so you have 600 rolls of a fair die. Tool: Google Sheets (free) or Excel. Check: Column A holds 600 whole numbers from 1 to 6, and every edit to the sheet rolls them all again.
2. In B1 type Running share of sixes. In B2 type =COUNTIF($A$2:A2,6)/(ROW()-1) and copy it down to B601. Then, in empty cells, type =COUNTIF(A2:A7,6), =COUNTIF(A2:A61,6), and =COUNTIF(A2:A601,6) to count the sixes after 6, 60, and 600 rolls. Before your next edit, copy all three counts and the matching shares into the first scaffold table, because the next edit rerolls the sheet. Tool: Google Sheets (free) or Excel. Check: Each running share is a decimal from 0 to 1; the share in B601 equals the 600-roll count divided by 600.
3. Select B1:B601 and use Insert, then Chart, and choose Line chart. Add a second column C with =1/6 in every row from C2 to C601 and include it in the chart, so a flat line marks the probability of a six. Save or sketch the chart before the sheet rerolls. Tool: Google Sheets (free) or Excel. Check: The line jumps around near the left edge and settles close to the flat line at 1 ⁄ 6, about 16.7%, near the right edge.
4. Open a second tab named Two dice. In A2 and B2 type =RANDBETWEEN(1,6), in C2 type =A2+B2, and copy all three down to row 361 for 360 rolls of two dice. Count the events with =COUNTIF(C2:C361,7), =COUNTIF(C2:C361,">=10"), and =COUNTIFS(A2:A361,6,B2:B361,6), and record them in the second scaffold table. Tool: Google Sheets (free) or Excel. Check: Column C holds sums from 2 to 12; the three counts are whole numbers, and the double-six count is the smallest.
5. Open a third tab named Penguins. In cell A1 type =IMPORTDATA("https://raw.githubusercontent.com/allisonhorst/palmerpenguins/main/inst/extdata/penguins.csv"). In Excel, save the file from that address, then use Data, then From Text/CSV, and load it starting in cell A1. Tool: Google Sheets (free) or Excel. Check: Row 1 shows the eight column names, and the penguins fill rows 2 through 345.
6. Fill the third scaffold table with =COUNTIFS(A2:A345,"Adelie",B2:B345,"Dream"), =COUNTIF(B2:B345,"Dream"), =COUNTIF(A2:A345,"Adelie"), =COUNTIFS(A2:A345,"Adelie",B2:B345,"Torgersen"), and =COUNTIF(B2:B345,"Torgersen"). Divide to get each probability, and format it to four decimals. Tool: Google Sheets (free) or Excel. Check: Every count is a whole number; the Adelie count on Dream is smaller than the Dream total, and the Torgersen counts match each other.

Fallback route

When to use it: Use this route if the penguin file will not import or no spreadsheet is available.

For the dice, roll a real die 60 times and two real dice 60 times by hand, tally the results on paper, and complete the first two scaffold tables with 6 and 60 rolls in place of the larger runs. For the penguins, work from the model data below, which reports the species-by-island counts from the real file. Answer every analysis question and the critique step from these records, and submit the same items for equal credit.

This route earns the same credit as the primary route.

These counts from the real file give you every penguin number the lab needs without importing it.

Fallback model data

Quantity from the file	Value
Penguins in the file	344
Adelie penguins	152 (44 on Biscoe, 56 on Dream, 52 on Torgersen)
Chinstrap penguins	68 (all on Dream)
Gentoo penguins	124 (all on Biscoe)
Penguins on each island	Biscoe 168, Dream 124, Torgersen 52
P(Adelie / Dream), to four decimals

Part 4 — The Scaffold

Record how the share of sixes behaves as the rolls pile up, so you can see the short run swing and the long run settle.

Student scaffold table 1

Number of rolls	Number of sixes	Share of sixes (percent, one decimal)	Gap from 16.7% (percentage points)
6
60
600

Compare the shares from your 360 rolls of two dice with the chances you count from the 36 equally likely outcomes.

Student scaffold table 2

Event	Count in your 360 rolls	Your share (percent, one decimal)	Chance from the 36-outcome list (fraction)
Sum = 7
Sum ≥ 10
Sum = 7 or sum ≥ 10
Both dice show 6

Count penguins by species and island so you can compare a conditional probability with the overall one.

Student scaffold table 3

Quantity	Count or probability (four decimals)
Adelie penguins on Dream
All penguins on Dream
P(Adelie / Dream)
All Adelie penguins
P(Adelie) across all 344 penguins
Adelie penguins on Torgersen
All penguins on Torgersen
P(Adelie / Torgersen)

Part 5 — Analysis Questions

1. Using your first table, how far was your share of sixes from 1 ⁄ 6, about 16.7%, after 6, 60, and 600 rolls? Which run came closest, and what does the law of large numbers say about why?
2. Describe your line chart in words: where the running share swings most, where it settles, and how close it ends to the flat line at 1 ⁄ 6. Did any stretch of rolls go without a six, and did the next roll become more likely to be a six?
3. In your second table, compare each of your shares with its chance from the 36-outcome list. Then add your counts for sum = 7 and sum ≥ 10 and compare the total with your count for the third row. Explain why they match, using the words mutually exclusive and the addition rule.
4. From your third table, compare P(Adelie | Dream) with P(Adelie), and then P(Adelie | Torgersen) with P(Adelie). Are species and island independent? Also explain why P(Dream | Adelie) is a different number from P(Adelie | Dream).
5. Choose one and answer it in 3–4 sentences. (a) A friend rolls a die six times, never gets a six, and decides the die is unfair: what can and cannot your simulation tell them? (b) The penguin probabilities come from one observational field log: what can and cannot they tell you about why a species lives on an island, or about penguins that were never logged?

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 5 Data Lab tables above. In the lab I simulated 600 rolls of a fair die and 360 rolls of two fair dice in a spreadsheet, and I used the Palmer penguins file, where Dream island holds 56 Adelie and 68 Chinstrap penguins, Torgersen holds 52 Adelie, and Biscoe holds 44 Adelie and 124 Gentoo. Please do five things. First, tell me how many sixes my 600 rolls should have produced and whether my run proves the die is fair. Second, describe the shape of the distribution of the sum of two fair dice and say whether it is skewed. Third, say whether these counts show that living on Torgersen makes a penguin an Adelie. Fourth, give the chance of getting exactly one six when two fair dice are rolled. Fifth, as a stretch question for later in the course, say what a poll's 95% confidence statement means.

Test the response for these likely failures:

• Chatbots often state an exact count as if it were guaranteed, saying 600 rolls give exactly 100 sixes, or they report a share for my run that I never gave them.
• Chatbots sometimes call the sum of two dice skewed toward the high or low totals, or name a skew by where most outcomes pile up.
• Chatbots often slide from a dependence in the table to a cause, saying Torgersen's conditions make penguins Adelie because every Torgersen penguin is Adelie.
• Chatbots often compute 1 ⁄ 6 × 5 ⁄ 6 = 5 ⁄ 36 for exactly one six and stop, forgetting that the six can be on the first die or the second.
• On the stretch question, chatbots often say a 95% confidence statement means a 95% chance that the true value is inside this one poll's interval.

Write 3–5 sentences naming one thing the chatbot got right and at least two places it went wrong or overreached, quoting its words and correcting each with evidence from your own tables, your 36-outcome list, or the penguin counts.

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: "After 600 rolls the die gave exactly 100 sixes, as it always must, so the die is perfectly fair. The sum of two dice is skewed to the right, because the big totals stretch out. Every penguin on Torgersen is an Adelie, so living on Torgersen turns penguins into Adelie. The chance of exactly one six on two dice is 1 ⁄ 6 × 5 ⁄ 6 = 5 ⁄ 36, and nothing more. A poll that is 95% confident means there is a 95% chance its one interval is right." Mark it up: quote each flawed sentence, name what is wrong (an invented or overclaimed statistic, a wrong skew claim, cause read from association, a dropped number of ways, or a misread confidence statement), cite the evidence from your tables, the 36-outcome list, or the penguin counts, and write a corrected sentence beneath it; for the last sentence, say what the course will check later. This route assesses the same objective (Objective 4 — apply the basic rules of probability, including conditional probability and independence, and compute and interpret the expected value and standard deviation of a discrete random variable, including binomial settings) 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

• All three completed scaffold tables
• Your line chart of the running share of sixes, or a labelled sketch of it
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Data Lab report on chance in the long run: three completed tables (the share of sixes after 6, 60, and 600 simulated rolls; event shares from 360 simulated rolls of two dice beside their counted chances; and penguin counts with conditional probabilities by island), a line chart of the running share of sixes or a labelled sketch of it, written answers to five analysis questions, and a critique of a chatbot's answers about the same chances, 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
Running and recording the simulation	The die and two-dice simulations, or the hand-rolled fallback, are run and recorded: the sixes are counted after 6, 60, and 600 rolls, the two-dice events are counted, and every share is the count divided by the number of rolls.	The simulations are run, but one or two counts or shares are missing or wrong, or a share is not divided by the right number of rolls.	No simulation results are recorded, or the counts are invented rather than produced.
Picturing the long run	A line chart of the running share of sixes, or a labelled sketch of it, shows the share against a flat line at 1 ⁄ 6 with labelled axes, and its description says where the share swings and where it settles.	The chart or sketch is present, but the reference line or an axis label is missing, or the description does not say where the share settles.	No chart, sketch, or description of the running share is submitted.
Checking chances and the addition rule	Analysis questions 1 to 3 are answered accurately: each gap from 1 ⁄ 6 is computed, the law of large numbers explains why the longest run tends to come closest, a streak is not treated as making a six due, and the sum = 7 or sum ≥ 10 count is matched to the addition rule for mutually exclusive events.	The questions are attempted, but one or two answers treat a streak as making a six due, expect an exact match to the counted chance, or leave out a reason.	Analysis questions 1 to 3 are missing or do not use the simulation results.
Conditional probability and independence	Analysis question 4 divides each conditional probability by the group it is given, compares P(Adelie / Dream) and P(Adelie / Torgersen) with P(Adelie), concludes that species and island are not independent, and explains why P(Dream / Adelie) differs.	The question is attempted, but one conditional probability uses the wrong denominator, or the verdict rests on one comparison without a reason.	Analysis question 4 is missing or does not use the penguin counts.
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, the 36-outcome list, or the penguin counts, 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 chance data can and cannot show	The chosen reflection option is answered in 3–4 sentences that name a specific limit, such as a short run proving nothing about fairness or an observational log showing association rather than cause.	The reflection is answered but stays general or does not name a specific limit.	No reflection is submitted.

Quality Gate

• Every count and share in the tables comes from your own simulation, your own tallies, your own formulas, or the fallback model data, never from memory or a chatbot.
• Each share is its count divided by the number of rolls, and the count for sum = 7 or sum ≥ 10 equals the two separate counts added together.
• The line chart shows the running share against a flat line at 1 ⁄ 6, with both axes labelled.
• Each conditional probability divides by the group it is given, and the independence verdict compares a conditional probability with the overall one.
• 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, the 36-outcome list, or the penguin counts.