{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Geometric Probability & Vectors",
    "author": "Staples Education",
    "description": "Probability as a ratio of lengths, angles and areas, and vector components, sums, scaling and resultants. Authored: these are the last two shelves of the course sequence and the tutoring sessions did not reach them.",
    "created": "2026-08-06T04:14:56.061Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "probability-vectors",
      "title": "Geometric Probability & Vectors",
      "blurb": "Probability as a ratio of lengths, angles and areas, and vectors as magnitude-and-direction — components, sums, scaling, and resultants.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "probability",
        "vectors"
      ],
      "questions": [
        {
          "id": "pvc-001",
          "topic": "Geometric Probability",
          "difficulty": "medium",
          "stem": "A circular target of radius 2 is painted at the centre of a square board of side 8. A dart lands at a uniformly random point on the board. What is the probability it lands inside the circle?",
          "choices": [
            "π/16",
            "π/4",
            "4π/8",
            "π/64"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Geometric probability is a ratio of measures: circle area over square area = π(2²)/8² = 4π/64 = π/16 ≈ 0.196.",
            "π/4 is the ratio for a circle INSCRIBED in the square — radius 4, not 2. This circle is much smaller than the board.",
            "4π/8 compares the circle's area to the square's SIDE rather than its area. An area must be divided by an area.",
            "π/64 uses the radius 2 without squaring it, or divides 4π by 64 incorrectly. 4π/64 reduces to π/16."
          ],
          "hint": "Favourable area over total area. Square both dimensions before dividing.",
          "diagram": "<svg viewBox=\"0 0 240 168\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"60,20 180,20 180,140 60,140\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><circle cx=\"120\" cy=\"80\" r=\"30\" fill=\"none\" stroke=\"#c0392b\" stroke-width=\"2\"/><line x1=\"120\" y1=\"80\" x2=\"150\" y2=\"80\" stroke=\"#c0392b\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"128\" y=\"76\" fill=\"#c0392b\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">2</text><text x=\"112\" y=\"158\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">8</text></svg>",
          "diagramCaption": "Uniform random point on the square. Not drawn to scale"
        },
        {
          "id": "pvc-002",
          "topic": "Geometric Probability",
          "difficulty": "easy",
          "stem": "A point is chosen at random on a segment from 0 to 10. What is the probability that it lies between 2 and 5?",
          "choices": [
            "3/10",
            "1/2",
            "7/10",
            "3/5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. On a one-dimensional region the measure is length: the favourable interval is 5 − 2 = 3 long, out of a total of 10, so the probability is 3/10.",
            "1/2 would be the answer for an interval of length 5, such as 0 to 5.",
            "7/10 is the probability of landing OUTSIDE the interval, 1 − 3/10.",
            "3/5 divides 3 by 5 rather than by the whole length 10."
          ],
          "hint": "Favourable length over total length. Watch that the interval's length is the difference, not its upper endpoint."
        },
        {
          "id": "pvc-003",
          "topic": "Geometric Probability",
          "difficulty": "medium",
          "stem": "A spinner is divided into sectors. What is the probability of landing in a sector with a central angle of 120°?",
          "choices": [
            "1/3",
            "1/2",
            "120/100",
            "2/3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. For a spinner the measure is angle: 120°/360° = 1/3.",
            "1/2 would need a 180° sector.",
            "120/100 is greater than 1, which no probability can be. The denominator is 360, not 100.",
            "2/3 is the probability of landing OUTSIDE that sector, 240°/360°."
          ],
          "hint": "The whole spinner is 360°. What fraction of it is this sector?"
        },
        {
          "id": "pvc-004",
          "topic": "Geometric Probability",
          "difficulty": "hard",
          "stem": "Two concentric circles have radii 3 and 5. A point is chosen at random inside the larger circle. What is the probability it falls in the ring between them?",
          "choices": [
            "16/25",
            "2/5",
            "9/25",
            "4/5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The ring's area is π(5²) − π(3²) = 25π − 9π = 16π, and the whole disc is 25π, so the probability is 16π/25π = 16/25 = 0.64.",
            "2/5 is the ratio of the RADII difference to the outer radius, (5−3)/5. Probability compares areas, not radii.",
            "9/25 is the probability of landing INSIDE the smaller circle, which is the complement of the answer.",
            "4/5 does not correspond to either region. Note that squaring the radii is what makes this ring bigger than a radius comparison suggests."
          ],
          "hint": "Subtract the inner area from the outer, then divide by the outer. The π cancels."
        },
        {
          "id": "pvc-005",
          "topic": "Vectors",
          "difficulty": "easy",
          "stem": "What is the magnitude of the vector ⟨3, 4⟩?",
          "choices": [
            "5",
            "7",
            "12",
            "25"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Magnitude is the Pythagorean length of the arrow: |⟨3, 4⟩| = √(3² + 4²) = √25 = 5.",
            "7 adds the components. They are perpendicular, so they combine by Pythagoras.",
            "12 multiplies them, which gives twice the area of the triangle they span — not a length.",
            "25 is the magnitude squared. Take the root."
          ],
          "hint": "The components are the legs of a right triangle; the magnitude is its hypotenuse.",
          "diagram": "<svg viewBox=\"0 0 170 148\" width=\"100%\" style=\"max-width:170px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"40\" y1=\"10\" x2=\"40\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"62\" y1=\"10\" x2=\"62\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"84\" y1=\"10\" x2=\"84\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"106\" y1=\"10\" x2=\"106\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"128\" y1=\"10\" x2=\"128\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"150\" y1=\"10\" x2=\"150\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"120\" x2=\"150\" y2=\"120\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"98\" x2=\"150\" y2=\"98\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"76\" x2=\"150\" y2=\"76\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"54\" x2=\"150\" y2=\"54\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"32\" x2=\"150\" y2=\"32\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"10\" x2=\"150\" y2=\"10\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"120\" x2=\"150\" y2=\"120\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"10\" x2=\"40\" y2=\"120\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"120\" x2=\"106\" y2=\"32\" stroke=\"#c0392b\" stroke-width=\"2.4\" stroke-linecap=\"round\"/><polygon points=\"106,32 96,42 112,44\" fill=\"#c0392b\"/><line x1=\"40\" y1=\"120\" x2=\"106\" y2=\"120\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"106\" y1=\"120\" x2=\"106\" y2=\"32\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"66\" y=\"138\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3</text><text x=\"114\" y=\"82\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4</text><text x=\"56\" y=\"76\" fill=\"#c0392b\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
          "diagramCaption": "The vector as an arrow from the origin. Not drawn to scale"
        },
        {
          "id": "pvc-006",
          "topic": "Vectors",
          "difficulty": "easy",
          "stem": "Add the vectors ⟨2, 3⟩ and ⟨5, −1⟩.",
          "choices": [
            "⟨7, 2⟩",
            "⟨7, 4⟩",
            "⟨10, −3⟩",
            "⟨−3, 4⟩"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Vectors add component by component: ⟨2 + 5, 3 + (−1)⟩ = ⟨7, 2⟩.",
            "⟨7, 4⟩ adds the y-components as 3 + 1, missing the negative sign on the second one.",
            "⟨10, −3⟩ multiplies the components instead of adding them.",
            "⟨−3, 4⟩ subtracts the first from the second rather than adding."
          ],
          "hint": "Add x to x and y to y, minding the sign on the −1."
        },
        {
          "id": "pvc-007",
          "topic": "Vectors",
          "difficulty": "medium",
          "stem": "A vector runs from the point (2, 3) to the point (7, 1). What is its component form?",
          "choices": [
            "⟨5, −2⟩",
            "⟨−5, 2⟩",
            "⟨9, 4⟩",
            "⟨5, 2⟩"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Subtract the tail from the head: ⟨7 − 2, 1 − 3⟩ = ⟨5, −2⟩. The negative y-component says the vector points downward.",
            "⟨−5, 2⟩ subtracts in the opposite order, giving the vector from (7, 1) back to (2, 3) — the same arrow reversed.",
            "⟨9, 4⟩ adds the coordinates instead of subtracting.",
            "⟨5, 2⟩ has the right x but drops the sign on y. Going from y = 3 to y = 1 is a decrease."
          ],
          "hint": "Head minus tail. The sign of each component tells you which way it points."
        },
        {
          "id": "pvc-008",
          "topic": "Vectors",
          "difficulty": "medium",
          "stem": "What is 3⟨2, −4⟩?",
          "choices": [
            "⟨6, −12⟩",
            "⟨5, −1⟩",
            "⟨6, −4⟩",
            "⟨2, −12⟩"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Scalar multiplication scales every component: ⟨3(2), 3(−4)⟩ = ⟨6, −12⟩. The direction is unchanged and the magnitude is tripled.",
            "⟨5, −1⟩ adds 3 to each component. Scalar multiplication multiplies; adding a number to a vector is not defined.",
            "⟨6, −4⟩ scales only the x-component, which would change the vector's direction as well as its length.",
            "⟨2, −12⟩ scales only the y-component, with the same problem."
          ],
          "hint": "Multiply both components. If only one changes, the arrow points somewhere new — which scaling must not do."
        },
        {
          "id": "pvc-009",
          "topic": "Vectors",
          "difficulty": "hard",
          "stem": "What is the direction angle of the vector ⟨1, √3⟩, measured counterclockwise from the positive x-axis?",
          "choices": [
            "60°",
            "30°",
            "45°",
            "120°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. tan θ = y/x = √3/1 = √3, so θ = 60°. The vector lies in the first quadrant, where the inverse tangent needs no adjustment. Its components 1 and √3 are the short and long legs of a 30-60-90 triangle.",
            "30° is the angle whose tangent is 1/√3 — the vector ⟨√3, 1⟩, with the components swapped.",
            "45° would need equal components, tan θ = 1.",
            "120° would put the vector in the second quadrant, which needs a negative x-component. Here x = 1 is positive."
          ],
          "hint": "Take the inverse tangent of y over x, then check which quadrant the signs put you in."
        },
        {
          "id": "pvc-010",
          "topic": "Vectors",
          "difficulty": "medium",
          "stem": "A boat heads due east at 12 km/h while a current pushes it due north at 5 km/h. What is its resultant speed?",
          "choices": [
            "13 km/h",
            "17 km/h",
            "7 km/h",
            "√17 km/h"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The two velocities are perpendicular, so they add as a right triangle: √(12² + 5²) = √169 = 13 km/h. The 5-12-13 triple.",
            "17 adds the speeds. That would only be right if both pushed in the SAME direction.",
            "7 subtracts them, which would apply if they were directly opposed.",
            "√17 adds the components before squaring. Each has to be squared first."
          ],
          "hint": "East and north are perpendicular. Draw the two arrows head to tail and measure the closing side."
        },
        {
          "id": "pvc-011",
          "topic": "Vectors",
          "difficulty": "medium",
          "stem": "Two vectors are described as equal. What must be true?",
          "choices": [
            "They have the same magnitude and the same direction",
            "They start at the same point",
            "They have the same magnitude only",
            "They lie on the same line"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A vector is defined by magnitude and direction alone — its position is not part of it. Two arrows anywhere on the plane are the same vector if they are the same length and point the same way.",
            "Starting point is exactly what a vector does NOT record. Translating an arrow leaves the vector unchanged, which is what makes vector addition position-independent.",
            "Same magnitude alone is not enough: ⟨3, 0⟩ and ⟨0, 3⟩ both have magnitude 3 and are not equal.",
            "Lying on the same line allows opposite directions — ⟨2, 0⟩ and ⟨−2, 0⟩ are collinear but not equal."
          ],
          "hint": "List everything a vector actually encodes. Is location one of them?"
        },
        {
          "id": "pvc-012",
          "topic": "Geometric Probability",
          "difficulty": "medium",
          "stem": "A 10 cm segment is cut at a random point. What is the probability that the shorter piece is less than 2 cm?",
          "choices": [
            "2/5",
            "1/5",
            "1/2",
            "4/5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The shorter piece is under 2 cm when the cut falls within 2 cm of EITHER end — that is the interval from 0 to 2 or from 8 to 10, a favourable length of 2 + 2 = 4 out of 10, giving 4/10 = 2/5.",
            "1/5 counts only one end. The condition is symmetric, so both ends qualify.",
            "1/2 would need a favourable length of 5, which corresponds to a different condition.",
            "4/5 is the complement region measured wrongly; the probability that the shorter piece is at least 2 cm is 6/10 = 3/5, not 4/5."
          ],
          "hint": "Two separate stretches of the segment satisfy the condition, one at each end. Add both before dividing."
        },
        {
          "id": "pvc-013",
          "topic": "Geometric Probability (1D)",
          "difficulty": "medium",
          "stem": "A point is chosen uniformly at random on a segment of length 12. What is the probability it lands within a sub-segment of length 3?",
          "choices": [
            "1/3",
            "1/4",
            "3",
            "1/12"
          ],
          "correctIndex": 1,
          "explanations": [
            "1/3 comes from dividing 12 by 3 and inverting incorrectly, or from using a sub-segment of length 4. Divide the favorable length by the total length as written.",
            "Correct. In continuous sample spaces the counting formula fails, so probability is measured rather than counted: P = length of favorable region / length of whole region = 3/12 = 1/4.",
            "3 is the favorable length itself. A probability must fall between 0 and 1, so any answer above 1 signals that the division was skipped.",
            "1/12 would be the probability for a favorable region of length 1. The favorable region here is 3 units long."
          ],
          "hint": "When outcomes are uncountable, swap counting for measuring. The formula keeps its shape but changes what it counts."
        },
        {
          "id": "pvc-014",
          "topic": "Probability Zero",
          "difficulty": "hard",
          "stem": "A point is chosen uniformly at random on a segment. What is the probability it lands on one specific pre-chosen point, and what does that value mean?",
          "choices": [
            "0, and that outcome is impossible",
            "0, but the outcome is still possible",
            "Undefined, since a point has no length",
            "1 divided by the number of points on the segment"
          ],
          "correctIndex": 1,
          "explanations": [
            "The probability is indeed 0, but 'impossible' is the wrong reading. Some point certainly gets chosen, so outcomes of probability zero do occur — this is the key conceptual break between finite and continuous sample spaces.",
            "Correct. A single point has length 0, so P = 0/ℓ(S) = 0. Yet the process always selects some point, so probability-zero events are not impossible ones. In continuous spaces, 'probability 0' means 'negligible in measure', not 'cannot happen'.",
            "It is well defined: the measure of a single point is 0, and dividing by the segment's positive length gives exactly 0. Nothing here is ill-posed.",
            "The number of points on a segment is uncountably infinite, so this expression is not a real number. It is precisely the breakdown of the counting formula that forces the shift to measuring lengths."
          ],
          "hint": "Compute the value first, then think carefully about whether 'probability zero' and 'impossible' really mean the same thing."
        },
        {
          "id": "pvc-015",
          "topic": "Complement Rule",
          "difficulty": "medium",
          "stem": "A fair coin is flipped 3 times. What is the probability of getting at least one head?",
          "choices": [
            "3/8",
            "1/2",
            "7/8",
            "1/8"
          ],
          "correctIndex": 2,
          "explanations": [
            "3/8 is the probability of exactly one head. 'At least one' covers exactly one, exactly two, and exactly three.",
            "1/2 is the probability of a head on a single flip. Across three flips the chance of seeing at least one is considerably higher.",
            "Correct. Use the complement: the only way to fail is all tails, with probability (1/2)³ = 1/8. So P(at least one head) = 1 − 1/8 = 7/8. The complement rule is the standard tool for 'at least one' problems precisely because it replaces three cases with one.",
            "1/8 is the probability of NO heads — the complement itself, not the event asked about."
          ],
          "hint": "'At least one' has many favorable cases but only one unfavorable case. Count the easy side and subtract."
        },
        {
          "id": "pvc-016",
          "topic": "Permutations vs Combinations",
          "difficulty": "medium",
          "stem": "From a class of 10 students, how many different 3-person committees can be formed?",
          "choices": [
            "720",
            "120",
            "30",
            "1000"
          ],
          "correctIndex": 1,
          "explanations": [
            "720 is ₁₀P₃ = 10!/7!, the number of ordered ARRANGEMENTS. A committee has no internal ranking, so the three orderings of the same trio should not be counted separately.",
            "Correct. A committee is unordered, so this is a combination: ₁₀C₃ = 10!/(3!·7!) = 120. Note that 720/3! = 120 — the permutation count divided by the 6 ways to order each committee.",
            "30 is 10 × 3, which multiplies the two given numbers directly. Counting problems of this kind need factorials, not a single product.",
            "1000 is 10³, which would count selections made WITH replacement and WITH order — allowing the same student to be picked three times."
          ],
          "hint": "Decide whether swapping two chosen people creates a different outcome. That single question separates the two formulas."
        },
        {
          "id": "pvc-017",
          "topic": "Vector Magnitude",
          "difficulty": "easy",
          "stem": "Find the magnitude of the vector ⟨5, −12⟩.",
          "choices": [
            "7",
            "13",
            "17",
            "√17"
          ],
          "correctIndex": 1,
          "explanations": [
            "7 subtracts the components, 12 − 5. Magnitude combines them through the Pythagorean Theorem, not by subtraction.",
            "Correct. The magnitude is √(5² + (−12)²) = √(25 + 144) = √169 = 13 — the 5-12-13 triple. Squaring removes the sign, so the negative component contributes exactly as a positive one would.",
            "17 adds the absolute values of the components, 5 + 12. That is the distance travelled along the axes, not the straight-line length of the vector.",
            "√17 adds the components before squaring rather than squaring each first. Each component is squared individually, then summed."
          ],
          "hint": "A vector's magnitude is the hypotenuse of the right triangle its components form. The sign of a component does not survive squaring."
        },
        {
          "id": "pvc-018",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "A student adds vectors (3, 4) and (1, 2) by adding their magnitudes: 5 + √5. Why is that wrong?",
          "choices": [
            "Vectors add COMPONENTWISE, giving (4, 6), whose magnitude is √52 — not the sum of the magnitudes.",
            "Magnitudes can never be added.",
            "The student should have multiplied the magnitudes.",
            "Nothing is wrong."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Vector addition is componentwise: (3+1, 4+2) = (4, 6), with magnitude √(16+36) = √52. Magnitudes only add when the vectors point in exactly the same direction.",
            "Magnitudes CAN be added — but only when the vectors are parallel and in the same direction, which these are not.",
            "Multiplication of magnitudes is part of the dot product, which produces a scalar rather than a resultant vector.",
            "Adding magnitudes overstates the resultant whenever the vectors differ in direction."
          ],
          "hint": "Direction matters. Two vectors pointing different ways partly cancel.",
          "kind": "reasoning"
        },
        {
          "id": "pvc-019",
          "topic": "Reasoning: Probability",
          "difficulty": "hard",
          "stem": "A dart lands at random on a square target with a circle inscribed in it. A student says the chance of hitting the circle is 1/2 \"because the circle covers about half\". What is the actual reasoning?",
          "choices": [
            "It is 1/2; the circle covers half the square.",
            "Geometric probability is a ratio of AREAS: πr² over (2r)², which is π/4 ≈ 0.785.",
            "It is 1/4, since the circle touches four sides.",
            "It cannot be found without the radius."
          ],
          "correctIndex": 1,
          "explanations": [
            "The estimate is well short. An inscribed circle covers considerably more than half its square.",
            "Correct. Geometric probability compares areas. With side 2r, the square has area 4r² and the circle πr², so the probability is π/4 ≈ 78.5%.",
            "The number of tangent points has nothing to do with the area covered.",
            "The radius cancels out, which is why the answer is the same for every square."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}