Week 5 — Does One Fact Change the Chance? A Short Probability Report

Assignment 5 of the term — count outcomes, apply the complement, addition, and multiplication rules, check two events for independence, and explain a conditional chance to someone new to statistics.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, evidence-based report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. Last week you read conditional distributions in a two-way table and learned that association is not causation; this week those same shares become conditional probabilities, and you test whether one fact changes the chance of another. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles; this assignment is where you put them together.

The task

Two fair dice land in 36 equally likely ordered outcomes, from (1, 1) to (6, 6). A common year has 12 months, and 7 of them have 31 days. A random guess on a four-option multiple-choice question picks the keyed option with probability 0.25. The Palmer penguins field log is a real observational dataset of 344 penguins; Biscoe holds 44 Adelie and 124 Gentoo (168 penguins), Dream holds 56 Adelie and 68 Chinstrap (124), and Torgersen holds 52 Adelie. From that table, P(the penguin is Gentoo | the penguin is on Biscoe) = 124 ⁄ 168 ≈ 0.7381, while P(Gentoo) = 124 ⁄ 344 ≈ 0.3605 across all 344 penguins.

Work these four problems, showing your arithmetic:

1. Two fair dice give 36 equally likely ordered outcomes, and 6 of them give a sum of 10 or more. How many of the 36 outcomes give a sum of 9 or less?
2. For two fair dice, the event "the first die shows 6" covers 6 of the 36 outcomes, the event "the sum is 10 or more" covers 6 of the 36 outcomes, and 3 outcomes are in both. How many of the 36 outcomes are in at least one of the two events?
3. A random guess on a four-option multiple-choice question picks the keyed option with probability 0.25. Two guesses on two different questions do not affect each other. What is the probability that both guesses pick the keyed option?
4. Given that the first of two fair dice shows 6, there are 6 equally likely outcomes, and 3 of them give a sum of 10 or more. What is P(sum ≥ 10 | first die 6)?

Beside each answer, add one brief interpretation: which rule you used, what the value means as a probability, and one use or limit of it. For problem 4, compare your answer with P(sum ≥ 10) = 1 ⁄ 6 and say whether the two events are independent; then run the same check on the penguin table, comparing P(Gentoo | Biscoe) ≈ 0.7381 with P(Gentoo) ≈ 0.3605.

Explain it to a non-expert

A volunteer guide at a nature center has never studied statistics. Visitors keep asking, "If I see a penguin on Biscoe Island, how likely is it to be a Gentoo, and does living on Biscoe make a penguin a Gentoo?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to the penguin counts, and names one limit — for example, that the field log is an observational study, so the link between island and species is an association rather than a cause, or that the chances describe the 344 penguins that were logged. Explain any technical term, such as "given" or "independent", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a library volunteer who notices that laptop loans seem to depend on the time of day. Model answer: "Of the laptops lent this month, most of the evening loans went to students with an exam that week, a much bigger share than across the whole day. So knowing it is evening changes the chance that the borrower has an exam coming up. That tells us the two go together in our records, but not that evenings cause exams, and one busy month may not hold for the whole year."

Use that pattern for the penguins, with your own explanation and evidence.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the volunteer before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about probability rules, conditional probability, and independence. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: two fair dice land in 36 equally likely ordered outcomes; a common year has 12 months, and 7 of them have 31 days; a random guess on a four-option question picks the keyed option with probability 0.25; the Palmer penguins field log, an observational dataset of 344 penguins, has 44 Adelie and 124 Gentoo on Biscoe (168), 56 Adelie and 68 Chinstrap on Dream (124), and 52 Adelie on Torgersen, so P(Gentoo | Biscoe) = 124/168 = 0.7381 and P(Gentoo) = 124/344 = 0.3605, each rounded to four decimals.
PROBLEM P1: Two fair dice give 36 equally likely ordered outcomes, and 6 of them give a sum of 10 or more. How many of the 36 outcomes give a sum of 9 or less?
PROBLEM P2: For two fair dice, the event "the first die shows 6" covers 6 of the 36 outcomes, the event "the sum is 10 or more" covers 6 of the 36 outcomes, and 3 outcomes are in both. How many of the 36 outcomes are in at least one of the two events?
PROBLEM P3: A random guess on a four-option multiple-choice question picks the keyed option with probability 0.25. Two guesses on two different questions do not affect each other. What is the probability that both guesses pick the keyed option?
PROBLEM P4: Given that the first of two fair dice shows 6, there are 6 equally likely outcomes, and 3 of them give a sum of 10 or more. What is P(sum ≥ 10 | first die 6)?
After the four answers, I must briefly interpret each one: which rule I used, what the value means as a probability, and one use or limit of it; for P4, compare my answer with P(sum >= 10) = 1/6, say whether the two events are independent, and run the same check on the penguin table. I must then write a 3-5 sentence message to a nature-center volunteer who is new to statistics, explaining how likely a Biscoe penguin is to be Gentoo, giving a reason tied to the penguin counts, and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct probability computations (40 points), rule choice and probability reasoning (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct count or probability = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation, such as a fraction instead of a decimal.
REASONING UNITS: For P1-P3 separately, an accurate account of which rule applies and what the value means plus a use or limit tied to the dice, calendar, or quiz setting = 2; exactly one of the two = 1; neither = 0. For P4, BOTH an accurate independence check that compares the conditional probability with the overall one for the dice and for the penguin table AND a final message with a data-specific reason and a named limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0; naming a rule alone does not earn a unit. Treating mutually exclusive events as independent, or swapping the condition in P(A | B), does not count as accurate. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to statistics, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message to the volunteer AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised probability report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised probability report with all four answers, their brief interpretations, and your message to the volunteer — written, or recorded as audio or video with a transcript, or as a labelled sketch of the outcome grid or the penguin table with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

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.

Equivalent route (no AI required)

This route assesses the same Week 5 skills — counting outcomes in a sample space, applying the complement, addition, and multiplication rules, computing a conditional probability, and checking two events for independence — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message to the volunteer. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised probability report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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