Week 5 — Does One Fact Change the Chance? A Short Probability Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned to read a probability as a long-run share, to list a sample space, to use the complement, addition, and multiplication rules, and to read a conditional probability off a two-way table. 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. In this assignment you work four short problems, interpret each one, and explain a conditional chance to someone who has never studied statistics. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single probability report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of which rule you used, what the value means, and one use or limit of it;
• your message to the volunteer, explaining the conditional chance with a reason tied to the penguin counts and naming one limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes 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. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the counting, the probabilities, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a spreadsheet for the arithmetic. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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