NCEA Level 3Organising informationAS 91585

Tables, trees and Venn diagrams from written information

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

Organise first, then calculate

  • Questions arrive as a paragraph of words.
  • A table, tree or Venn makes them easy.
  • Skipping this step: a 2025 finding.
Key idea

Let the wording choose

  • Two variables: a two-way table.
  • One, the other, or both: a Venn diagram.
  • "Of those who": a tree.
InformationStructure
Two variablestwo-way table
Overlapping groupsVenn diagram
"Of those who..."tree
Values of Xprobability table
Worked example

Words into a two-way table

  • 200 orders, 120 takeaway.
  • 95 with food, 45 of them takeaway.
  • Place the givens, mark the gaps.
FoodNo foodTotal
Takeaway45?120
Dine-in???
Total95?200
Worked example

Fill the gaps by subtraction

  • Totals first.
  • Each gap: total − known.
  • Check the last cell twice.
  1. dine-in: 200 − 120 = 80row totals
  2. no food: 200 − 95 = 105column totals
  3. takeaway, no food: 120 − 45 = 75
  4. dine-in, food: 95 − 45 = 50
  5. dine-in, no food: 80 − 50 = 30
  6. check: 105 − 75 = 30
Worked example

The finished table

  • Rows and columns add up.
  • Now ready for any probability.
FoodNo foodTotal
Takeaway4575120
Dine-in503080
Total95105200
Worked example

A Venn diagram, middle first

  • P(sport) = 0.55, P(music) = 0.3.
  • Both: 0.12, in the middle.
  • Then the other three regions.
A Venn diagram, middle first
Worked example

Each region, worked

  • Circle minus the middle.
  • Neither: 1 minus the rest.
  1. only sport = 0.55 − 0.12 = 0.43
  2. only music = 0.3 − 0.12 = 0.18
  3. neither = 1 − (0.43 + 0.12 + 0.18)
  4. = 1 − 0.73 = 0.27
  5. 0.43 + 0.12 + 0.18 + 0.27 = 1
Worked example

Given "or" instead of "both"

  • Use the addition rule.
  • Neither = 1 − P(S or M).
  1. P(S or M) = P(S) + P(M)addition rule
  2. − P(S and M)
  3. 0.73 = 0.55 + 0.3 − P(S and M)
  4. P(S and M) = 0.55 + 0.3 − 0.73
  5. = 0.12
  6. neither = 1 − 0.73 = 0.27outside both circles
Worked example

Completing the Venn: a 2025 error

  • Wrong: circles as "only".
  • Take the middle away first.
  1. only sport = 0.55whole circle
  2. only music = 0.3whole circle
  3. neither = 1 − (0.55 + 0.12 + 0.3)
  4. = 1 − 0.97 = 0.03overlap counted 3 times
  5. right: 0.43, 0.12, 0.18, 0.27
Worked example

The same numbers as a table

  • Four Venn regions, four table cells.
  • Totals match the circles.
  • Use whichever reads more easily.
MusicNo musicTotal
Sport0.120.430.55
No sport0.180.270.45
Total0.30.71
Worked example

Words into a tree

  • First stage: rural 0.3, urban 0.7.
  • "40% of rural" goes after rural.
  • "10% of urban" goes after urban.
Words into a tree
Worked example

Multiply along, add across

  • Multiply along each path.
  • Add the paths you want.
  1. rural, tank: 0.3 × 0.4 = 0.12
  2. rural, none: 0.3 × 0.6 = 0.18
  3. urban, tank: 0.7 × 0.1 = 0.07
  4. urban, none: 0.7 × 0.9 = 0.63
  5. 0.12 + 0.18 + 0.07 + 0.63 = 1
  6. P(tank) = 0.12 + 0.07two tank paths
  7. = 0.19
Worked example

A probability table sums to 1

  • X: people in a car.
  • Missing: 1 minus the rest.
  1. P(X = 1) = 0.45, P(X = 2) = 0.3
  2. P(X = 4) = 0.1
  3. P(X = 3) = p
  4. 0.45 + 0.3 + p + 0.1 = 1the table sums to 1
  5. 0.85 + p = 1
  6. p = 1 − 0.85 = 0.15
  7. every value from 0 to 1
Worked example

The completed table

  • The gap is 0.15.
  • Every value from 0 to 1.
  • Together they total 1.
xP(X = x)
10.45
20.3
3? = 0.15
40.1
Key idea

Where the marks are

  • Achieved: pick a structure, find a probability.
  • Merit: complete Venns and trees accurately.
  • Excellence: use it to judge a claim.
Summary

What to take away

  • Organise the words first, then check every set totals 1.
Two variablestwo-way table
OverlapsVenn, middle first
"Of those"second-stage branch
Every settotals 1 (or n)
Missing P1 minus the rest
Checkeach p from 0 to 1
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Draw the structure first.
  • Show your working.
GivenFind
1A 0.6, B 0.5, both 0.3Venn regions
2wet 0.3; late 0.5, 0.2P(late)
30.15, 0.4, ?, 0.1P(X = 2)
440: 25 N, 18 H, 8 bothP(neither)
Show answer

1. Only A 0.6 − 0.3 = 0.3, both 0.3, only B 0.5 − 0.3 = 0.2, neither 1 − 0.8 = 0.2. 2. 0.3 × 0.5 + 0.7 × 0.2 = 0.15 + 0.14 = 0.29. 3. P(X = 2) = 1 − (0.15 + 0.4 + 0.1) = 0.35. 4. Only netball 17, both 8, only hockey 10, neither 5; P(neither) = 5/40 = 0.125.

Worked example

The answers: questions 1 and 2

  1. 1. only A = 0.6 − 0.3 = 0.3
  2. only B = 0.5 − 0.3 = 0.2
  3. neither = 1 − (0.3 + 0.3 + 0.2)
  4. = 1 − 0.8 = 0.2
  5. 2. wet, late: 0.3 × 0.5 = 0.15
  6. dry, late: 0.7 × 0.2 = 0.14
  7. P(late) = 0.15 + 0.14 = 0.29
Worked example

The answers: questions 3 and 4

  1. 3. 0.15 + 0.4 + p + 0.1 = 1the table sums to 1
  2. 0.65 + p = 1
  3. P(X = 2) = 1 − 0.65 = 0.35
  4. 4. only N = 25 − 8 = 17
  5. only H = 18 − 8 = 10
  6. neither = 40 − (17 + 8 + 10)
  7. = 40 − 35 = 5
  8. P(neither) = 5 ÷ 40 = 0.125

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