Sample spaces, events, and counting
Assistance allowed only after prediction and observed evidence are recorded.
Prerequisites
Complete Day 24 (linear algebra wrap-up); if blocked, review Day 21 before continuing.
Learning Objective
Define a sample space, enumerate equally likely outcomes, and compute event probabilities by counting under explicitly stated fairness and independence assumptions.
Instructions
Study source: within this activity's existing reading/practice time, use Stat 110: Strategic Practice and Homework Problems, Strategic Practice and Homework 1; linked problem and solution document, then explain one connection to the local task in your own words. Keep the local exercise and acceptance checks; use its original explanation if the external source is unavailable. A probability model assumes a sample space $\Omega$ of equally likely outcomes and assigns $P(A) = |A|/|\Omega|$ to an event $A$. For two fair and independent six-sided dice, the sample space is the 36 ordered pairs $(a,b)$, each with probability $1/36$; fair marginals alone are insufficient: if the dice always match, the sum-7 probability is 0 and the at-least-one-6 probability is $1/6$. Required work: predict the counts first, then enumerate the pairs whose sum is 7 and the pairs containing at least one 6 (use inclusion-exclusion so $(6,6)$ is not double-counted). Optional work: repeat the counting exercise with three coins only after the required evidence is complete.
Practice & Prediction Log
For two fair and independent six-sided dice (so each ordered pair has probability $1/36$), predict then compute: (a) $P(\text{sum}=7)$; (b) $P(\text{at least one die shows }6)$. Give each answer as a fraction of the 36 ordered pairs and as a decimal, and state the fairness and independence assumptions you relied on.
Sum-7 ordered pairs are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). For at least one 6: 6 pairs have first die 6, 6 have second die 6, minus the double-counted (6,6).
(a) $P(\text{sum}=7) = 6/36 = 1/6 \approx 0.1667$. (b) $P(\text{at least one }6) = 11/36 \approx 0.3056$ (inclusion-exclusion: $6+6-1=11$). Both verified by enumerating all 36 ordered pairs under the stated fair-and-independent-dice assumption.
Your sample space is the 36 ordered pairs; both probabilities are derived by counting under the explicitly stated assumptions that each die is fair and the dice are independent (equivalently, each pair has probability 1/36), not recalled.
Authoritative Source Resources & Provenance
Stat 110: Strategic Practice and Homework Problems
Relevant Section / Range: Strategic Practice and Homework 1; linked problem and solution document
Verified Version & Date: Public edition checked 2026-11-01; moving pages may change
Concept-organized questions plus solutions supply probability teaching practice missing from reference notes. Use early sets, not an entire semester or an unverified free textbook.
Counting, fractions and the Day 25 probability definitions; use only the identified early problems.
Use the original localized explanations, exercises and checks already present in this activity; no external account or purchase is required for that fallback.
Compared Candidate Sources & Selection Analysis (2)
Retained for accessible definitions and a full statistics sequence; Stat 110 adds targeted probability practice.
Comparable introductory statistics text; remains a supplementary reference for estimation rather than replacing concept-targeted probability practice.
Introductory Statistics 2e
Relevant Section / Range: Chapter 3: Probability Topics (sample spaces, events, equally likely outcomes)
Verified Version & Date: 2e web edition (Verified: 2026-10-30)
Peer-reviewed open textbook with beginner-grade rigor for sample spaces and counting-based probability.
High-school algebra
All required definitions, counts, and worked answers are reproduced locally in this lesson.
Compared Candidate Sources & Selection Analysis (3)
Rigorous but paced for a full graduate semester; exceeds the depth needed for Days 25-27 training-objective foundations.
Accessible video-first coverage; noncommercial/share-alike terms and video pacing make it a supplementary rather than primary link.
Strong intuition-building course; the printed textbook is paywalled, so it complements rather than replaces the openly licensed OpenStax text.
CS229 Review of Probability Theory
Relevant Section / Range: Section 2: Probability spaces, events, and axioms
Verified Version & Date: 2020 public course notes (Verified: 2026-10-30)
Compact machine-learning-oriented probability review connecting axioms to model assumptions.
Multivariate calculus notation exposure
Self-contained definitions and derivations in activity instructions and offline guide.
Compared Candidate Sources & Selection Analysis (2)
Co-selected: provides the gentler beginner narrative; CS229 provides the ML-oriented notation bridge.
Strong full-course treatment with noncommercial/share-alike terms; heavier than the compact CS229 reference for this three-day block.