Random events

Consider a random experiment. The set $S$ of all possible outcomes is called the sample space of the experiment. Thus, if the experiment has $m$ possible outcomes denoted by $o_1, ..., o_m$ , then the sample space is

S=\{ o_1, ..., o_m\}

Any subset $E\subset S$ is called an event. Let's perform the experiment. We say that event E has occurred if the outcome that occurred is in $E$ .

Some special events

The opposite event of an event $E$ is the event $E^\prime$ . Clearly, if $E$ occurs, $E^\prime$ cannot occur, and vice versa.
The sure event is the event $E=S$ . This name makes sense, as this event occurs every time.
The impossible event is the event $E=\{ \}$ . Again, this name makes sense, as this event will never occur (the experiment always produces an outcome).
The atomic events are the events containing exactly one outcome, that is, the atomic events are $\{o_1\}, \{o_2\}, ..., \{o_m\}$ . Often we will not distinguish between outcomes and atomic events, and simply write $o_1, ..., o_m$ as well.

Example 1

A coin is flipped. The sample space is
$S=\{H,T\}$
A die is rolled. The sample space is
$S=\{1,2,3,4,5,6\}$
- The "event $E=\{2,4,6\}$ occurs" can also expressed as "an even number occurs".
- The opposite event is $E^\prime=\{1,3,5\}$ , that is, the event that "an odd number occurs".
- That "a number smaller than $3$ was observed" can also be expressed as "the event $E=\{1,2\}$ occurred".
A coin is flipped twice. The sample space is
$S=\{HH,HT, TH, TT\}$
where we mean with $HH$ that $H$ occurred in the first flip, and $H$ in the second flip. And so on.
- The observation that "exactly one head occurred" can also be expressed as "the event $E=\{HT, TH\}$ occurred".
- The event "no head occurred" can also be expressed as "the event $E=\{TT\}$ occurred".
- The opposite event is $E^\prime = \{HT, TH, HH\}$ , that is, the event that at least one head occurs.
Random selection of a person in a group of $m$ people. The sample space consists of all possible people in the group:
$S=\{p_1, p_2, ... , p_m\}$
An event could be $E$ ="person has green eyes". This event $E$ contains then all persons with green eyes.

Exercise 1

Determine the sample space and the number of outcomes for the following random experiments:

Flipping a coin three times.
Rolling a die twice.

Solution

$S=\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}, \vert S \vert = 8$ .
$S=\{11,12,13,14,15,16, ..., 61, 62, 63, 64, 65, 66\}, \vert S \vert = 36$ . Note that with " $15$ " we mean a " $1$ " in the first roll and a " $5$ " in the second roll, and so on.

Exercise 2

Express each event as a selection of outcomes.

You roll a die twice.

$E$ ="the sum of the two observed numbers is even"

$F$ ="at least one 6 occurred"

$G$ ="no six occurred"

$H$ ="the sum is between $6$ and $8$ (including $6$ and $8$ )"
You randomly select a number between $1$ and $10$ from a box

$I$ ="the selected number is a prime"

$J$ ="the selected number is divisible by $3$ and bigger than $5$ "

Solution

$E=\{11,13,15,22,24,26,31,33,35,42,44,46,51,53,55,62,64,66\}$

$F=\{16,26,36,46,56,61,62,63,64,65,66\}$

$G=F^\prime$

$H=\{15,16,24,25,26,33,34,35,42,43,44,51,52,53,61,62\}$

$I=\{2,3,5,7\}$

$J=\{6,9\}$