The two-way table and conditional probability

Statistics and probability, grade 10 (4 units) Lesson 15 of 20Learning

The two-way table and conditional probability

The two-way table

A two-way table sorts the data by two variables at once: one variable in the rows and one in the columns. Each cell is the intersection of two events.

All the residents of a neighbourhood were asked whether they want a cinema built in the shopping centre:

	Want one	Do not want one	Total
Young	75	85	160
Adults	140	50	190
Total	215	135	350

A resident of the neighbourhood is chosen at random.
$\displaystyle Y$ – the resident is young; $\displaystyle C$ – the resident wants a cinema built.

$\displaystyle {P(Y)}={\frac{160}{350}}={\frac{16}{35}}$ – from the row’s total.
$\displaystyle {P(C)}={\frac{215}{350}}={\frac{43}{70}}$ – from the column’s total.
$\displaystyle {P(Y\cap C)}={\frac{75}{350}}={\frac{3}{14}}$ – from the cell where the row and the column meet.

A table of probabilities

A table can hold probabilities instead of numbers of people. In such a table the four inner cells add up to 1, and the total of a row or a column is the probability of its event.

Given: $\displaystyle {P(A)}={0.6}$ , $\displaystyle {P(B)}={0.3}$ , $\displaystyle {P(A\cap B)}={0.2}$ .
The table is completed by subtracting, row by row and column by column:

	$\displaystyle B$	$\displaystyle \overline{B}$	Total
$\displaystyle A$	0.2	$\displaystyle {0.6-0.2}={0.4}$	0.6
$\displaystyle \overline{A}$	$\displaystyle {0.3-0.2}={0.1}$	$\displaystyle {0.4-0.1}={0.3}$	$\displaystyle {1-0.6}={0.4}$
Total	0.3	$\displaystyle {1-0.3}={0.7}$	1

From the table we read, for example, $\displaystyle {P(\overline{A}\cap\overline{B})}={0.3}$ : neither $\displaystyle A$ nor $\displaystyle B$ .

Conditional probability from the table

A conditional probability is the probability of an event when another event is already known to have happened. The probability of $\displaystyle C$ given $\displaystyle Y$ is written $\displaystyle P(C|Y)$ (also $\displaystyle P(C/Y)$ ).

You met a young resident of the neighbourhood. What is the probability that he wants a cinema?

The resident is known to be young, so the choice is only among the 160 young people – the young row is now the whole sample space.
75 of them want one.

P(C|Y)=\frac{75}{160}=\frac{15}{32}

Of all the residents $\displaystyle \frac{43}{70}\approx0.61$ want one, and of the young only $\displaystyle \frac{15}{32}\approx0.47$ .

The definition of conditional probability

In a table of counts the cell is divided by the row’s total. Dividing both by the number of residents, 350, gives the same fraction – and that is the definition of conditional probability:

\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}

In the neighbourhood: $\displaystyle {P(Y\cap C)}={\frac{75}{350}}$ and $\displaystyle {P(Y)}={\frac{160}{350}}$ .

P(C|Y)=\frac{\frac{75}{350}}{\frac{160}{350}}=\frac{75}{160}=\frac{15}{32}

From the formula, multiplying by $\displaystyle P(B)$ : $\displaystyle {P(A\cap B)}={P(B)\cdot P(A|B)}$ .

The difference between

\displaystyle P(A|B)

and

\displaystyle P(B|A)

$\displaystyle P(A|B)$ and $\displaystyle P(B|A)$ are two different questions: their numerators are the same intersection, but the denominators differ – once $\displaystyle P(B)$ and once $\displaystyle P(A)$ .

The residents’ committee held a meeting for all 135 residents who do not want a cinema. $\displaystyle \overline{C}$ – the resident does not want one.

You met a resident at the meeting. The probability that he is young: $\displaystyle {P(Y|\overline{C})}={\frac{85}{135}}={\frac{17}{27}}$ – among the 135 who do not want one.
You met a young resident. The probability that he does not want one: $\displaystyle {P(\overline{C}|Y)}={\frac{85}{160}}={\frac{17}{32}}$ – among the 160 young.

The words “given that …” say what to divide by: the event known to have happened is the denominator.

Exercise 1

A school asked its 400 students in years 10, 11 and 12 whether they come to school by bus:

	Year 10	Year 11	Year 12	Total
By bus	60	45	35	140
Not by bus	90	75	95	260
Total	150	120	130	400

A student is chosen at random.

What is the probability that the student comes by bus?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power