The two-way table

A two-way table sorts data by two variables at once: one variable in the rows and one in the columns. Each inner cell is the intersection of two events, and the total row and the total column hold each event on its own.

A school asked its 450 students in years 10, 11 and 12 whether they volunteer in the community. This is a $\displaystyle 3\times2$ table – three rows and two columns of data:

A student is chosen at random. $\displaystyle V$ – the student volunteers; $\displaystyle T$ – the student is in year 12.

• $\displaystyle {P(V)}={\frac{180}{450}}={\frac{2}{5}}$ – from the column’s total.
• $\displaystyle {P(T\cap V)}={\frac{72}{450}}={\frac{4}{25}}$ – from the cell where the row and the column meet.
• $\displaystyle {P(\overline{V})}={\frac{270}{450}}={\frac{3}{5}}$ – the second column, and also $\displaystyle 1-\frac{2}{5}$.
• A union – “in year 12 or volunteers”: $\displaystyle {P(T\cup V)}={\frac{140+180-72}{450}}={\frac{248}{450}}={\frac{124}{225}}$. The 72 are in both the row and the column, so they are taken away once.

Dividing each cell by the number of all the students, 450, gives a table of probabilities. In such a table:

• the inner cells add up to 1;
• the total of a row or a column is the probability of its event;
• in each row and each column, the inner cells add up to its total.

Many questions give only some of the cells, as percentages of the whole population. The rest is completed by subtraction, row by row and column by column.

In a certain town 40% of the residents are under 30.
25% of the residents use the bike-sharing service.
15% of the residents are under 30 and use the service.
$\displaystyle Y$ – the resident is under 30; $\displaystyle S$ – the resident uses the service.

From the table: the probability that a resident chosen at random is 30 or older and does not use the service is $\displaystyle {P(\overline{Y}\cap\overline{S})}={0.5}$.

When what is given is a link between two cells, one cell is written $\displaystyle x$, the other cells are expressed with it, and an equation is solved.

In a youth club 45% of the members are girls.
30% of the members play chess.
The probability that a member chosen at random is a boy who does not play chess is 3 times the probability that the member is a girl who plays chess.
$\displaystyle G$ – a girl; $\displaystyle C$ – plays chess.

The solution below writes $\displaystyle {P(G\cap C)}={x}$, completes the table with $\displaystyle x$, and writes an equation from the link between the cells.

The probability that a member is a girl who plays chess is $\displaystyle 0.125$, and that a member is a boy who does not play – $\displaystyle 0.375$.

A gym has 300 members:

A member is chosen at random.

What is the probability that the member trains in the morning?

What is the probability that the member is a woman who trains in the evening?

What is the probability that the member is a man or trains in the morning?

$\displaystyle {P(A)}={0.3}$, $\displaystyle {P(B)}={0.55}$, $\displaystyle {P(\overline{A}\cap\overline{B})}={0.25}$.
Build a two-way table of probabilities, and fill in all its cells.

What is $\displaystyle P(A\cap B)$?

What is $\displaystyle P(A\cap\overline{B})$?

A shop sells the phones of three companies: 25% of the phones sold are company A’s, 45% company B’s, and the rest company C’s.
10% of all the phones sold are company A’s and were sold with insurance.
18% of all the phones are company B’s and were sold with insurance.
12% of all the phones are company C’s and were sold with insurance.
A phone that was sold is chosen at random. Build a $\displaystyle 3\times2$ two-way table.

What is the probability that the phone was sold with insurance?

What is the probability that the phone is company C’s and was sold without insurance?

30% of the candidates in a college entrance exam came from the north, and the rest from the centre.
65% of the candidates passed the exam.
The probability of choosing at random a candidate from the centre who failed is 1.5 times the probability of choosing a candidate from the north who passed.

Write $\displaystyle x$ for the probability that a candidate is from the north and passed, and find $\displaystyle x$.

What is the probability that a candidate is from the north and failed?