# Two-Way Tables

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## Two-Way Tables

Two-way tables show information about two characteristics, one being presented in the columns and one presented in the rows.

For example, a table may show the amount of boys and girls who preferred football or rugby.

You will need to able to construct two-tables as well as use them to calculate probabilities of specific events occurring.

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## Understanding Two-Way Tables

The following two-way table shows information about the hair colour of $100$ children:

What do the values mean?

Examples:

There are $\textcolor{blue}{10}$ boys with black hair.

There are $\textcolor{green}{18}$ children with blonde hair.

There are $\textcolor{pink}{55}$ girls.

There are $\textcolor{orange}{100}$ children overall.

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## Completing Two-Way Tables

You may be asked to complete incomplete two-way tables.

The following two-way table shows information about the hair colour of another $100$ children, with some missing values:

In this table, we need to find the missing values, $\textcolor{red}{a}$ and $\textcolor{blue}{b}$:

$\textcolor{red}{a}$: we know there are $36$ children with brown hair, and $21$ of these are boys, so we can find how many girls have brown hair:

$36-21=15\\$ $\textcolor{red}{a} = 15$

$\textcolor{blue}{b}$: we know there are $100$ children overall, and $51$ of these are boys, so we can find how many girls there are overall:

$100-51=49\\$ $\textcolor{blue}{b} = 49$

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## Finding Probabilities from Two-Way Tables

Two-way tables also allows us to calculate probabilities.

The following two-way table shows information about how $50$ people travel to school or work:

Examples of calculating probability:

• One person from this table is chosen at random. Calculate the probability of this person being under $25$ years old and walking to school or work.

From the table, we can see $11$ under $25$‘s walk to school or work, out of a possible $50$ people.

Therefore, the probability is $\dfrac{11}{50}$.

• One person from this table is chosen at random. Calculate the probability of this person travelling by bus to school or work.

From the table, we can see $7$ people in total take the bus, out of a possible $50$ people.

Therefore, the probability is $\dfrac{7}{50}$.

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## Conditional Probabilities from Two-Way Tables

We can also calculate conditional probabilities from two-way tables.

The following two-way table shows information about how $50$ people travel to school or work:

Conditional probability example:

• One person from this table is chosen at random. Given that the person chosen is over $25$, calculate the probability of this person driving in a car to school or work.

We are given that the person is over $25$, and as there are $25$ people over this age, this is the total number of possible people that can be chosen.

From the table, we can see $12$ over $25$‘s drive to school or work.

This gives us a probability of $12$ out of a total $25$ people:

$\dfrac{12}{25}$

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## Example 1: Probability

The following table shows the favourite colours of Year $7$ and Year $8$ students.

A student is chosen at random. Find the probability of this student’s favourite colour being red.

[2 marks]

From the table we can see that $11$ students’ favourite colour is red, out of a possible $30$:

Probability $= \dfrac{11}{30}$

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## Example 2: Conditional Probability

The following table shows the favourite colours of different Year $7$ and Year $8$ students.

A year $8$ student is chosen at random. Find the probability of this student’s favourite colour being green.

[3 marks]

We are given that the student is in Year $8$, and as there are $14$ students in this year, this is the total number of possible people that can be chosen.

From the table, we can see $3$ Year $8$ students chose green as their favourite colour.

This gives us a probability of $3$ out of a total $14$ people:

$\dfrac{3}{14}$

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## Two-Way Tables Example Questions

Question 1: The following table shows some primary and secondary school students’ favourite subjects. Find the missing values $a, b$ and $c$.

[2 marks]

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$a$: $39-13=26$

$b$: $45-14=31$

$c$: $8+2=10$

Gold Standard Education

Question 2: The following table shows some people’s favourite sports.

A random person is chosen from the group. Calculate the probability of the person chosen’s favourite sport being tennis.

[2 marks]

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Total: $40$ people

People with favourite sport as tennis: $11$

Probability: $\dfrac{11}{40}$

Gold Standard Education

Question 3: The following table shows some students favourite art activity.

Given that a girl is chosen at random, calculate the probability of the girl’s favourite art form being painting.

[3 marks]

Level 8-9GCSE AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

Total girls: $32$

Girls with favourite art as painting: $6$

Probability: $\dfrac{6}{32}$

This could be simplified down to $\dfrac{3}{16}$