Sample spaces, Venn diagrams, tree diagrams and contingency tables – Counting Techniques (Senior 6 Basic Mathematics) | Ganzaa

Counting Techniques

Lessons

Senior 6 · Basic Mathematics

Counting Techniques

5 lessons

Lesson 1 of 5

Sample spaces, Venn diagrams, tree diagrams and contingency tables

25 min

Counting before calculating

How many ways can a class choose a captain and a vice-captain? How many different lunches can a canteen serve from its menu? How many students in a school play neither football nor volleyball? Each question asks for a number of possibilities, and each can be answered by listing, if the list is short. This unit builds methods for when the list is too long to write out. It starts with four ways of organising the outcomes so that none is missed and none is counted twice.

Definition

An experiment is an action with a set of possible results, such as rolling a die. Each result is an outcome. The set of all outcomes is the sample space $S$ . An event is a subset of $S$ . For any set $A$ , $n (A)$ is the number of elements in $A$ .

Example: rolling a die, $S = \{1, 2, 3, 4, 5, 6\}$ . The event "an even number" is $E = \{2, 4, 6\}$ , so $n (E) = 3$ .

1. Venn diagrams

In an S6 class of 40 students, 22 play football, 15 play volleyball and 6 play both. How many play neither?

Fill the diagram from the inside out: first the overlap (6), then "football only" $22 - 6 = 16$ and "volleyball only" $15 - 6 = 9$ . The students who play at least one sport number $16 + 6 + 9 = 31$ , so $40 - 31 = 9$ play neither.

The same reasoning gives the inclusion–exclusion rule. Adding $n (F)$ and $n (V)$ counts the overlap twice, so it is subtracted once:

n(F \cup V) = n(F) + n(V) - n(F \cap V) = 22 + 15 - 6 = 31

Worked example

Worked example 1: three sets. Of 50 students, 20 are in the Debate club (D), 18 in Drama (R) and 15 in Music (M). 7 are in both D and R, 5 in both D and M, 4 in both R and M, and 2 are in all three. How many are in no club, and how many in exactly one?

Inside out. All three: 2. Exactly D and R: $7 - 2 = 5$ . Exactly D and M: $5 - 2 = 3$ . Exactly R and M: $4 - 2 = 2$ .

D only: $20 - 5 - 3 - 2 = 10$ . R only: $18 - 5 - 2 - 2 = 9$ . M only: $15 - 3 - 2 - 2 = 8$ .

At least one club: $10 + 9 + 8 + 5 + 3 + 2 + 2 = 39$ . No club: $50 - 39 = 11$ . Exactly one club: $10 + 9 + 8 = 27$ .

Check with inclusion–exclusion for three sets:

n(D \cup R \cup M) = 20 + 18 + 15 - 7 - 5 - 4 + 2 = 39

This agrees with the count above.

2. Tree diagrams

A school canteen serves rice or cooked bananas, with beans, peas or fish. A tree diagram shows every possible plate: one branch for each choice at each stage. Each path from the start to the right-hand end is one outcome.

There are 6 paths, so 6 possible plates. Notice that $6 = 2 \times 3$ : every one of the 2 first choices can be followed by each of the 3 second choices. The next lesson turns this into a rule.

3. Contingency tables

A contingency (two-way) table counts individuals classified by two characteristics at once. The 120 S6 students of a school (illustrative):

	Boarding	Day	Total
Sciences	30	18	48
Languages	14	16	30
Humanities	22	20	42
Total	66	54	120

Each cell is an "and": 16 students are Languages and day. The totals let you check your reading: rows and columns must add up to 120.

The Organic Chemistry Tutor solves word problems with three overlapping categories. Notice that he always starts with the centre region, as in worked example 1.

Worked example

Worked example 2. From the contingency table, how many students are in Sciences or boarding (or both)?

Sciences: 48. Boarding: 66. The 30 who are both have been counted twice, so

n(\text{Sciences} \cup \text{Boarding}) = 48 + 66 - 30 = 84

Check by counting the cells directly: $30 + 18 + 14 + 22 = 84$ . ✓

Warning

Filling a Venn diagram from the outside in is the most common error. "22 play football" includes the 6 who play both, so writing 22 in the "football only" region counts those 6 twice. Always start from the centre.

Tip

Choose the tool by the question: Venn diagram for overlapping groups, tree diagram for choices made in stages, contingency table for two ways of classifying the same individuals.

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Practice

A coin is tossed and a die is rolled. List the sample space. How many outcomes are there?
In a group of 50 people, 30 drink tea, 25 drink coffee and 10 drink both. How many drink neither?
$n(A \cup B) = 40$ , $n (A) = 25$ and $n (B) = 22$ . Find $n(A \cap B)$ .
Use a tree diagram to list the outcomes for the sexes of the children in a family of three. How many outcomes have exactly two girls?
From the contingency table in this lesson, how many students are day students not in Sciences?
In worked example 1, how many students are in exactly two clubs?
In a survey of 100 farmers, 60 grow maize, 45 grow beans and 15 grow neither. How many grow both? Explain your method.

Show answersHide answers

H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6: 12 outcomes.
$n(T \cup C) = 30 + 25 - 10 = 45$ , so $50 - 45 = 5$ drink neither.
$40 = 25 + 22 - n(A \cap B)$ , so $n(A \cap B) = 7$ .
BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG: 8 outcomes; exactly two girls: BGG, GBG, GGB, so 3.
$16 + 20 = 36$ .
$5 + 3 + 2 = 10$ .
$100 - 15 = 85$ grow at least one, and $85 = 60 + 45 - x$ , so $x = 20$ grow both.

Important

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$ . Fill Venn diagrams from the centre outwards, and check every table with its totals.

Sources and credits

Text, diagrams, data, worked examples and exercises: Written by Ganzaa. All data are illustrative.

Video: Venn Diagram Word Problems With 3 Categories, The Organic Chemistry Tutor, embedded from YouTube.

1Counting before calculating

21. Venn diagrams

32. Tree diagrams

43. Contingency tables

Counting before calculating

1. Venn diagrams

2. Tree diagrams

3. Contingency tables