MATH U113 · Doubt clinic · index
Doubts, worked from zero
One card per doubt you send, filed on a topic page that matches the notes. Twenty-two cards so far, all Module 1: three from the 13 Sep lecture slides, thirteen from the 14 Sep problem set and two more slides, and six Devore worked examples (2.22, 2.26–2.29, 2.35, 2.36) filed 15 Sep.
Every doubt you send gets one card: the question as set, where the idea lives in the notes, the explanation rebuilt from first principles with the picture drawn, the mistake the question is designed to catch, and a twin to try with the card closed. Cards live on topic pages — one page per idea, matching the notes — so revising a topic means one page, not a hunt. The table below lists every card; filter it by topic, module or the date you sent it. Recurring doubt, answered once: when to use the addition rule and when to use combinations (also at the top of the counting page).
Topic pages
Sets, complements and the addition rule
King or ace · even first die or total 8 · P(A ∪ Bᶜ) from three numbers.
Module 1 · §2.3Counting with equally likely outcomes
Poker hand · three hats · dinosaurs in six packets · phones in service order · widget inspector.
Module 1 · §2.4Conditional probability, the multiplication rule and trees
Three magazine columns (Devore 2.26) · blood typing (2.27–2.28) · DVD brands and repairs (2.29).
Module 1 · §2.5Independence, reliability diagrams, repeated trials
Four-component system · pairwise vs mutual on two dice · the circuit with Bayes on failure · three vehicles.
Module 1 · §2.4Total probability and Bayes, with a tree
Four boxes · ball moved and moved back · three director candidates · the strike and two-bags slides.
All cards
| Card | The question, in one line | Topic page | Source | Filed |
|---|---|---|---|---|
| A king or an ace | One card from a shuffled deck: P(king or ace)? Why no overlap term here but one in the dice example. | Sets & the addition rule | Lecture slide 17 | 13 Sep |
| Even first die or total 8 | Two dice: P(even on the first die or a total of 8)? Where 18, 5 and 3 come from and why 3 is subtracted. | Sets & the addition rule | Lecture slide 17 | 13 Sep |
| 2 aces and 3 jacks in a poker hand | Five-card hand: P(2 aces and 3 jacks)? Why combinations, why multiply, why C(52,5) underneath. | Counting & equally likely outcomes | Lecture slide | 13 Sep |
| P(A ∪ Bᶜ) from three numbers | P(A) = P(B) = ½, P(Aᶜ ∩ Bᶜ) = ⅓: find P(A ∪ Bᶜ) by filling the four Venn regions. | Sets & the addition rule | Problem set Q3 | 14 Sep |
| Three hats, nobody gets his own | Three men take hats at random: P(no one gets his own) — a derangement, listed. | Counting & equally likely outcomes | Problem set Q5 | 14 Sep |
| Six cereal packets, three favourite dinosaurs | Five equally likely models per packet: P(all three favourites appear in six packets) — inclusion–exclusion on "missing". | Counting & equally likely outcomes | Problem set Q7 | 14 Sep |
| Fifteen phones in a random service order | All cordless in the first ten; only two types left after ten (the hint decoded); two of each type in the first six. | Counting & equally likely outcomes | Problem set Q8 | 14 Sep |
| Widget inspector: at least 9 tests | 3 defectives among 12, tested one by one: P(at least 9 tests), P(at least 10) — position of the last defective. | Counting & equally likely outcomes | Problem set Q11 | 14 Sep |
| Four-component system; C fails given it works | A and B and (C or D): P(system works), and P(C fails | system works) — why it is not 0.2. | Independence & reliability | Problem set Q9 | 14 Sep |
| Pairwise vs mutual independence on two dice | Red 3, green 4, total 7: pairwise independent (yes) but not mutually (no) — checked on the grid. | Independence & reliability | Problem set Q10 | 14 Sep |
| The circuit: operates, and which device is faulty | Three parallel branches with failure probabilities: P(operates), P(operates | C or E fails), P(C or E faulty | dead). | Independence & reliability | Problem set Q12 | 14 Sep |
| Three vehicles, 70% pass | All pass, at least one fails, exactly one passes, at most one, all three given at least one — eight sequences. | Independence & reliability | Problem set Q15 | 14 Sep |
| Four boxes, one component drawn | Boxes of different sizes and defect rates, a box chosen at random: P(defective) — the size decoy. | Total probability & Bayes | Problem set Q13 | 14 Sep |
| Ball moved to box 2 and back | P(red then red); P(box 1 ends as it began) — the eleven-ball denominator. | Total probability & Bayes | Problem set Q14 | 14 Sep |
| Three director candidates, new syllabus | Chances 4 : 2 : 3, syllabus probabilities 0.3, 0.5, 0.8: P(new syllabus); P(Z | new syllabus). | Total probability & Bayes | Problem set Q18 | 14 Sep |
| Two slides: the strike; the two bags | P(completed on time) = 0.488 and P(black from bag 2) = 38/63 — why the products add. | Total probability & Bayes | Lecture slides | 14 Sep |
| iPod shuffle: first Beatles song is the fifth played | 100 songs, 10 Beatles, random order: (90·89·88·87·10)/(100·99·98·97·96) = .0679, by permutations and by combinations. | Counting & equally likely outcomes | Devore Ex. 2.22 | 15 Sep |
| Three magazine columns: four conditional probabilities | Fill the eight Venn regions from seven numbers, then P(A | B), P(A | B ∪ C), P(A | at least one), P(A ∪ B | C). | Conditional probability | Devore Ex. 2.26 | 15 Sep |
| Blood typing: "at least three must be typed" | Why that event is "first two are not O+"; (2/3)(3/4) = ½; and P(third is O+) = ¼ (Ex. 2.28). | Conditional probability | Devore Ex. 2.27–2.28 | 15 Sep |
| Three DVD brands and warranty repairs | Tree: P(brand 1 and repair) = .125, P(repair) = .205, P(brand 1 | repair) = .61 — Bayes without the name. | Conditional probability | Devore Ex. 2.29 | 15 Sep |
| Two suppliers' batches: why three numbers multiply | P(two batches pass on a random day) = (.8)(.9)(.4) = .288 — the hidden "two batches arrived" condition. | Independence & reliability | Devore Ex. 2.35 | 15 Sep |
| Solar arrays: series-parallel vs total-cross-tied | Two chains of three in parallel: .927 (two routes agree); ties at every column: .970, and why ties help. | Independence & reliability | Devore Ex. 2.36 | 15 Sep |
What the cards share
- "Equally likely" is what licenses counting. Well shuffled, fair dice, "at random" — spot the phrase, then every probability is favourable ÷ total.
- "Or" = union = add, then subtract the overlap — and the overlap is zero exactly when the two events cannot co-occur. Draw the sample space (a grid, a list) and you can see the overlap instead of guessing it.
- Total probability is a tree — cases that cannot overlap and cover everything, one product per route, add the routes; Bayes is one route divided by the sum (Q13, Q14, Q18, slides, quiz Q1).
- Independence is a computation, and it does not survive conditioning on a system built from the event — pairwise ≠ mutual (Q10); P(C fails | system works) ≠ P(C fails) (Q9, Q12).
- "And" inside a selection = multiply the choices, all in the same mode (unordered hands over unordered hands). Count the cards the question names; if they fill the hand, there is no further factor.
"Devore §2.2 only: give me four 'A or B' questions on one card or two dice, two where the events are mutually exclusive and two where they overlap, and make me say which before I compute." · "Devore §2.3: check my counting for 'exactly 2 aces in a 5-card hand' and tell me whether my numerator and denominator are in the same mode." Stay clear of poker hand rankings in general — only the counting pattern is examinable.