Two Way Tables

What are two-way tables?

Two-way tables allow us to consider two characteristics within a set of data
- For example, we may be interested in the number of students studying Spanish or German
  - We may also be interested in how many of those students are in year 12 and how many are in year 13
- Spanish/German would be one characteristic in the two-way table, year 12/13 would be the second
One of the characteristics will be represented by the columns, the other by the rows
A two-way table should include row totals and column totals
- The row/column totals are sometimes called marginal (or sub-) totals
- Where the row totals and column totals meet, we have the grand total
- Marginal totals can be really useful in two-way table questions
  - If they're not mentioned, or not included in a given table, add them in!

Once a two-way table is completed, with marginal totals, the values within it can be used to determine probabilities

How do I draw and complete a two-way table?

To construct a two-way table from information given in words in a question
- identify the two characteristics
- use rows for one characteristic and columns for the other
- add an extra row and column for the marginal totals (and grand total)
Work your way through each sentence in the question
- fill in any values you can directly from the information given
- be prepared to come back to any information that cannot be put into the two-way table directly
  - some information may need combining in order to deduce a value

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

This is a matter of going from the words used in the question to probability phrases
- Aim to rephrase the question in your head using AND and/or OR statements
- e.g. The probability of selecting a year 12 student who studies German is P("year 12 AND German")
Once you are clear what parts of the two-way table are required you can begin to write down the probability
- the numerator will be taken from the main body of the two-way table
  - for "year 12 AND German" this would be the cell in the table where the row/column for "year 12" meets the row/column for "German"
- the denominator will be the total of the group we are choosing from
  - this could be either a row/column (marginal) total or the grand total
  - if we are choosing from the whole group it would be the grand total
  - if we are choosing from just year 12 students, say, it would be the total for the year 12 row/column

How do I work with two-way tables and conditional probability?

Probabilities that only involve a subset of the things in a two-way table are called conditional probabilities
- For example with male and female students studying various languages, you might want to know the probability that a male student studies German
- This would require considering only the 'male' row or column of the two-way table when finding the probability
Conditional probability questions are often (but not always!) introduced by the expression 'given that...'
- For example 'Find the probability that a randomly chosen student studies German, given that the student is male'
- The answer would be the number of 'male AND German' students divided by the total for the male row or column
Conditional probabilities are sometimes written using the 'straight bar' notation $P (A | B)$
- That is read as 'the probability of A given B'
- For example $P (German | male)$ would be the probability that a student studies German, given that the student is male

CP Notes fig5, downloadable IGCSE & GCSE Maths revision notes

Examiner Tip

Work carefully when completing a two-way table
- double check your values add up to each row/column total
- check your totals add up to the grand total
If there are errors in your table, your probabilities will be incorrect and you could lose several marks

Worked example

At an art group children are allowed to choose between four activities; colouring, painting, clay modelling and sketching.
There is a total of 60 children attending the art group. 12 of the boys chose the activity colouring.
A total of 20 children chose painting and a total of 15 chose clay modelling. 13 girls chose clay modelling.
8 of the 30 boys chose sketching, as did 4 of the girls.

Construct a two-way table to show this information.

Construct the table carefully, remember to include marginal totals for the rows and columns.
Work through each sentence in turn, placing a value in the table where possible and coming back later to a sentence if need be.
Once those values are in place, work your way around the rest of the table until it is complete.
If you find you can't complete the table, look back at the question for some information you may have missed.

	Colouring	Painting	Clay modelling	Sketching	Total
Boys	12	30 - 12 - 2 - 8 = 8	15 - 13 = 2	8	30
Girls	30 - 12 - 13 - 4 = 1	20 - 8 = 12	13	4	60 - 30 = 30
Total	12 + 1 = 13	20	15	8 + 4 = 12	60

So the final two-way table is

	Colouring	Painting	Clay modelling	Sketching	Total
Boys	12	8	2	8	30
Girls	1	12	13	4	30
Total	13	20	15	12	60

You can do a quick check of your table values by ensuring the marginal totals add up to the grand total.

Find the probability that a randomly selected child

chose colouring,

ii)

is a boy who chose sketching.

For this part of the question we are not interested in whether the child is a boy or a girl.
So we will need the values from the (marginal) total column for colouring, 13.
There are 60 children in total.

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

ii)

This time we do need to consider the characteristic boy/girl as well as the activity sketching.
The value in the cell where 'boy' meets 'sketching' is 8. There are 60 children to select from.

$P (boy and colouring) = \frac{8}{60} (= \frac{2}{15})$

A girl is selected at random. Find the probability they chose the activity painting.

As we are only selecting from the girls, this will be "out of" 30 rather than the total of 60 that are in the group.
12 girls chose painting.

$P (painting) = \frac{12}{30} (= \frac{2}{5})$

Two Way Tables (Cambridge O Level Maths)

Revision Note

Two Way Tables

What are two-way tables?

How do I draw and complete a two-way table?

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

How do I work with two-way tables and conditional probability?

Examiner Tip

Worked example

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Author: Paul