Two-Way Tables (Cambridge International IGCSE Mathematics)

Two Way Tables

Two-way tables

What are two-way tables?

• Two-way tables are tables that compare two types of characteristics

  • For example, a college of 55 students has two year groups (Year 12 and Year 13) and two language options (Spanish and German)
  • The two-way table is shown: | | Spanish | German | |---|---|---| | Year 12 | 15 | 10 | | Year 13 | 5 | 25 |

How do I find probabilities from a two-way table?

• Draw in the totals of each row and column

  • Include an overall total in the bottom-right corner

    • It should be the sum of the totals above, or to its left (both work)

  • For the example above: | | Spanish | German | Total | |---|---|---|---| | Year 12 | 15 | 10 | 25 | | Year 13 | 5 | 25 | 30 | | Total | 20 | 35 | 55 |

• Use this to answer probability questions

  • If a random student is selected from the whole college, it will be out of 55

    • The probability a student selected from the college studies Spanish and is in Year 12 is $\frac{15}{55}$
    • The probability a student selected from the college studies Spanish is $\frac{20}{55}$

  • If a random student is selected from a specific category, the denominator will be that category total

    • The probability a student selected from Year 13 studies Spanish is $\frac{5}{30}$

Exam Hint Check your row and column totals add up to the overall total, otherwise all your probabilities will be wrong!

Worked Example At an art group, children are allowed to choose between colouring, painting, clay modelling and sketching.

A total of 60 children attend and are split into two classes: class A and class B. 12 of class A chose the activity colouring and 13 of class B chose clay modelling. A total of 20 children chose painting and a total of 15 chose clay modelling. 8 of the 30 children in class A chose sketching, as did 4 children in class B.

(a) Construct a two-way table to show this information.

Answer:

Read through each sentence and fill in the numbers that are given

* *	Colouring	Painting	Clay modelling	Sketching	Total
Class A	12	* *	* *	8	30
Class B	* *	* *	13	4	* *
Total	* *	20	15	* *	60

Use the row and column totals to fill in any obvious missing numbers

	Colouring	Painting	Clay modelling	Sketching	Total
Class A	12	* *	15 - 13 = 2	8	30
Class B	* *	* *	13	4	60 - 30 = 30
Total	* *	20	15	8 + 4 = 12	60

Use the row and column totals again to find the last few numbers

* *	Colouring	Painting	Clay modelling	Sketching	Total
Class A	12	30 - 12 - 2 - 8 = 8	2	8	30
Class B	30 - 12 - 13 - 4 = 1	20 - 8 = 12	13	4	30
Total	12 + 1 = 13	20	15	12	60

Write out your final answer

Final answer:	Final answer: Colouring	Final answer: Painting	Final answer: Clay modelling	Final answer: Sketching	Final answer: Total
Final answer: Class A	Final answer: 12	Final answer: 8	Final answer: 2	Final answer: 8	Final answer: 30
Final answer: Class B	Final answer: 1	Final answer: 12	Final answer: 13	Final answer: 4	Final answer: 30
Final answer: Total	Final answer: 13	Final answer: 20	Final answer: 15	Final answer: 12	Final answer: 60

(b) Find the probability that a randomly selected child

(i) chose colouring,

(ii) is in class A, who chose sketching.

Answer:

(i)

We are not interested in whether the child is in class A or B A total of 13 children chose colouring, out of 60 children

Final answer: P(colouring) =$\frac{13}{60}$

(ii)* *

8 children in class A chose sketching There are 60 children to select from

Final answer: P(class A and sketching) =$\frac{8}{60}=\frac{2}{15}$

(c) A child in class B is selected at random. Find the probability they chose painting.

Answer:

As we are only selecting from class B, this will be out of 30 (rather than the total of 60) 12 in class B chose painting

Final answer: P(painting, from class B only) =$\frac{12}{30}=\frac{2}{5}$