Operations on events and the Venn diagram

Probability, grade 11 (5 units) Lesson 2 of 16Learning

Operations on events and the Venn diagram

The union and the intersection

Two events make new events:

The union $\displaystyle A\cup B$ – “ $\displaystyle A$ or $\displaystyle B$ ”: the outcomes in at least one of them, those in both included.
The intersection $\displaystyle A\cap B$ – “ $\displaystyle A$ and $\displaystyle B$ ”: the outcomes in both.

In a Venn diagram the rectangle is the sample space, and each circle is an event. The part the two circles share is the intersection, and the two circles together are the union.

A jar holds 12 slips of paper, numbered from 1 to 12. One slip is taken out at random.
$\displaystyle A$ – “the number is divisible by 3”; $\displaystyle B$ – “the number is even”.

$\displaystyle {A\cap B}={\{6,12\}}$ , so $\displaystyle {P(A\cap B)}={\frac{2}{12}}={\frac{1}{6}}$ .
$\displaystyle {A\cup B}={\{2,3,4,6,8,9,10,12\}}$ , so $\displaystyle {P(A\cup B)}={\frac{8}{12}}={\frac{2}{3}}$ .

The complement

The complement of $\displaystyle A$ , written $\displaystyle \overline{A}$ , is all the outcomes not in $\displaystyle A$ : “ $\displaystyle A$ does not happen”. In a Venn diagram it is everything in the rectangle outside the circle of $\displaystyle A$ .

With the complement every part of a Venn diagram can be described. In the jar of 12 slips:

only $\displaystyle A$ – “ $\displaystyle A$ and not $\displaystyle B$ ”: $\displaystyle {A\cap\overline{B}}={\{3,9\}}$ .
only $\displaystyle B$ : $\displaystyle {\overline{A}\cap B}={\{2,4,8,10\}}$ .
neither $\displaystyle A$ nor $\displaystyle B$ – outside both circles: $\displaystyle {\overline{A}\cap\overline{B}}={\{1,5,7,11\}}$ . This is exactly the complement of the union: $\displaystyle {\overline{A\cup B}}={\overline{A}\cap\overline{B}}$ .

Disjoint events

Two events are disjoint when they have no outcome in common: $\displaystyle {A\cap B}={\emptyset}$ . They cannot happen together, and in a Venn diagram their circles do not touch.

In the jar of 12 slips, “the number is 1” and “the number is even” are disjoint.
$\displaystyle A$ and $\displaystyle \overline{A}$ are always disjoint, and their union is the whole sample space: $\displaystyle {A\cup\overline{A}}={U}$ .
“The number is less than 4” and “the number is greater than 9” are disjoint but not complementary: 5, for example, is in neither of them.

Numbers in a Venn diagram

A Venn diagram also sorts the data of a survey: each part holds how many people are in it. Fill it in from the inside out – the intersection first.

A class has 32 students.
18 students study music, 13 study drama, and 6 study both.
$\displaystyle M$ – the student studies music; $\displaystyle D$ – the student studies drama.

In the intersection: 6.
Only music: $\displaystyle {18-6}={12}$ . Only drama: $\displaystyle {13-6}={7}$ .
Neither music nor drama: $\displaystyle {32-(12+6+7)}={7}$ .

A student is chosen at random:

$\displaystyle {P(M\cup D)}={\frac{12+6+7}{32}}={\frac{25}{32}}$ ;
“studies exactly one of the two” – only music or only drama: $\displaystyle {\frac{12+7}{32}}={\frac{19}{32}}$ .

Exercise 1

A jar holds 20 slips of paper, numbered from 1 to 20.
One slip is taken out at random.
$\displaystyle A$ – “the number is divisible by 4”; $\displaystyle B$ – “the number is greater than 14”.

Write the events $\displaystyle A\cap B$ , $\displaystyle A\cup B$ and $\displaystyle A\cap\overline{B}$ as sets of outcomes.

Work out $\displaystyle P(A\cup B)$ .

/ – 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