Organising information: 7 exam questions solved step by step

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What this lesson covers

  • Most probability questions become easy once the information is in a table, a tree or a Venn diagram.
  • Choose the structure that matches how the information is given.
Key idea

7 questions: organise, then solve

  • 1. Two variables: a two-way table.
  • 2. Overlaps: a Venn, middle first.
  • 3. "Of those": a tree.
  • 4. Every set totals 1 (or n).
Your turn

Question 1 · Achieved, in context

  • 300 campground guests: 180 from overseas,
  • 210 booked online (the rest by phone),
  • 140 were overseas and booked online.
  • Find P(a guest is from NZ, by phone).
Worked example

Q1: words into a table

  • Two variables: where from, how booked.
  • Place the givens, mark the gaps.
OnlinePhoneTotal
Overseas140?180
NZ???
Total210?300
Worked example

Q1: fill the gaps

  • Each gap: total − known.
  • Check one cell twice.
  1. NZ: 300 − 180 = 120row totals
  2. phone: 300 − 210 = 90column totals
  3. overseas, phone: 180 − 140 = 40
  4. NZ, online: 210 − 140 = 70
  5. NZ, phone: 120 − 70 = 50
  6. check: 90 − 40 = 50
  7. P(NZ and phone) = 50 ÷ 300
  8. = 0.16674 d.p.
Your turn

Question 2 · Achieved

  • X: the number of phones in a household.
  • Find p, then P(X ≥ 3).
  • Careful: "at least 3" includes 3.
xP(X = x)
10.1
20.3
3p
40.2
50.05
Worked example

Q2: the table sums to 1

  • Missing: 1 minus the rest.
  • ≥ 3 means 3, 4 or 5.
  1. 0.1 + 0.3 + p + 0.2 + 0.05 = 1
  2. 0.65 + p = 1
  3. p = 1 − 0.65 = 0.35
  4. every value from 0 to 1
  5. P(X ≥ 3) = 0.35 + 0.2 + 0.05
  6. = 0.6
  7. P(X > 3) = 0.25leaves out X = 3
Your turn

Question 3 · Merit

  • Year 13 students at a Whangārei school:
  • 40% have a part-time job, 45% have
  • a driver licence, 15% have both.
  • Find P(exactly one of the two).
Worked example

Q3: each region

  • Both goes in the middle.
  • Circle minus the middle.
  1. both = 0.15the middle
  2. only job = 0.4 − 0.15 = 0.25
  3. only licence = 0.45 − 0.15 = 0.3
  4. neither = 1 − (0.25 + 0.15 + 0.3)
  5. = 1 − 0.7 = 0.3
  6. exactly one = 0.25 + 0.3only job or only licence
  7. = 0.55
  8. 0.4 + 0.45 = 0.85counts both twice
Worked example

Q3: the finished Venn

  • Middle first, then each circle.
  • Four regions, total 1.
Q3: the finished Venn
Your turn

Question 4 · Merit

  • Visitors to a Rotorua park: 60% see
  • the geysers, 45% the mud pools, and
  • 80% see at least one of them.
  • Complete the Venn diagram and table.
  • Careful: 80% is "or", not "both".
Worked example

Q4: both from "or"

  • Addition rule first.
  • Then the regions.
  1. P(G or M) = P(G) + P(M)addition rule
  2. − P(G and M)
  3. 0.8 = 0.6 + 0.45 − P(G and M)
  4. P(G and M) = 1.05 − 0.8 = 0.25
  5. only G = 0.6 − 0.25 = 0.35
  6. only M = 0.45 − 0.25 = 0.2
  7. neither = 1 − 0.8 = 0.2outside both
  8. 0.35 + 0.25 + 0.2 + 0.2 = 1
Worked example

Q4: the same regions as a table

  • Four regions, four cells.
  • Totals match P(G) and P(M).
Mud poolsNo mud poolsTotal
Geysers0.250.350.6
No geysers0.20.20.4
Total0.450.551
Your turn

Question 5 · Merit, in context

  • At a ski field, 70% of visitors hire gear.
  • Of those who hire, 15% book a lesson.
  • Of the rest, 5% book a lesson.
  • Expected lessons from 2000 visitors?
Worked example

Q5: words into a tree

  • First: hire 0.7, own gear 0.3.
  • "Of those who hire": after hire.
  • "Of the rest": after own.
Q5: words into a tree
Worked example

Q5: multiply along, add across

  • Two lesson paths.
  • Then times 2000.
  1. hire, lesson: 0.7 × 0.15 = 0.105
  2. own, lesson: 0.3 × 0.05 = 0.015
  3. P(lesson) = 0.105 + 0.015both lesson paths
  4. = 0.12
  5. (0.15 + 0.05) ÷ 2 = 0.1ignores 70 : 30
  6. expected = 2000 × 0.12
  7. = 240
  8. about 240 lessons that day
Your turn

Question 6 · Excellence

  • 50 Year 13s: 22 take Chemistry and
  • 18 take Physics. Three times as many
  • take neither as take both.
  • Find P(a student takes exactly one).
Worked example

Q6: an unknown in the middle

  • Let both = x.
  • Four regions total 50.
  1. let x = number taking both
  2. only C = 22 − x, only P = 18 − x
  3. neither = 3x
  4. (22 − x) + x + (18 − x) + 3x = 50
  5. 40 + 2x = 50
  6. x = 52x = 10
  7. only C 17, only P 13, neither 15
  8. P(exactly one) = (17 + 13) ÷ 50
  9. = 30 ÷ 50 = 0.6
Worked example

Q6: check with the Venn

  • Circles: 17 + 5 = 22, 5 + 13 = 18.
  • Regions total 50.
Q6: check with the Venn
Your turn

Question 7 · Excellence

  • X: the number of days a student's bus
  • is late in a school week.
  • Find k, then P(X ≥ 2).
x0123
P(X = x)0.3kk²0.14
Worked example

Q7: a quadratic in k

  • Sums to 1: an equation.
  • A probability is never < 0.
  1. 0.3 + k + k² + 0.14 = 1the table sums to 1
  2. k² + k − 0.56 = 0collect, then − 1
  3. (k − 0.4)(k + 1.4) = 0−0.4 × 1.4 = −0.56
  4. k = 0.4 or k = −1.4
  5. k = −1.4a probability is ≥ 0
  6. 0.3, 0.4, 0.16, 0.14total 1
  7. P(X ≥ 2) = 0.16 + 0.14
  8. = 0.3
Summary

The structures that solved all seven

  • Organise the words first, then check every set totals 1.
StepHow
Two-waygaps: total − known
Vennmiddle first, then the rest
Tree"of those" is a 2nd branch
P tablesums to 1, each from 0 to 1
Unknownform an equation, then check
Checkevery set totals 1 or n

More in AS 91585

Tables, trees and Venn diagrams from written information6 minConditional probability from tables, trees and Venn diagrams6 minIndependent or mutually exclusive? The two tests6 minRelative risk: times as likely, and when you need a ratio6 minTrue probability, model estimates and experimental estimates7 minOrganising information: 7 exam questions solved step by step11 minConditional probability: 7 exam questions solved step by step10 minIndependent or exclusive: 7 exam questions solved step by step10 minRelative risk: 7 exam questions solved step by step10 minProbability estimates: 7 exam questions solved step by step9 min

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