Essentials of Data Science – Probability and Statistical Inference - Sample Space and Events

In the previous note in Essentials of Data Science – Probability and Statistical Inference – Introduction, we have seen the need for a mathematical foundation for Data Science. In the note, we will learn basic terminologies of Probability Theory which will be helpful to specify and represent probabilistic models formulation for Data Science.

Table of Contents hide

1 Experiment (or a random experiment)

2 Sample Space and Events

2.1 Events

2.2 Complementary Event

Experiment (or a random experiment)

Any activity for which the outcome is uncertain can be thought of as an experiment or a random experiment. The uncertainty concern in the sense that the outcome of the experiment is not known until the experiment is completed. The outcome is not predictable with certainty in advance. But total possible outcomes are certain. For example,

Drawing a card from a deck to observe which card is drawn.
Tossing a coin to observe what turns up – Head or Tail.
Writing any entrance examination.
Drug test.

Sample Space and Events

The outcome of the experiment is not known in advance, but we assume that all the possible outcomes of the experiment are known. The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $\Omega$ .

Events

Any subset $E$ of the sample space is known as an event. So an event is a set of possible outcomes of the experiment. If the outcome of the experiment is contained in $E$ , then we say that $E$ has occured.

The selection of sample space, events are entirely dependent on the experiment and goals. The below examples show how to select sample space, events, and all possible outcomes.

Example 1:

If the outcome of an experiment consists in the determination of the gender of a newly born child, then $\Omega$ = $\{M,F\}$ . where $M$ and $F$ indicates Male and Female child, respectively. Suppose if $E$ = ${M}$ , then $E$ is the event that the child is a male. Similarly, if $E$ = ${F}$ , then $E$ is the event that the child is a female.

Example 2:

Suppose there are three student participating in a game. Each of the students may acquire a position in the game, the positions may be first, second or third. The sample space $\Omega$ consisting with {(1,2,3), (1,3,2), (2,1,3), (3,1,2), (3,2,1), (2,3,1),(2,2,1)}. The outcome (2,3,1) means first student gets position 2, second student gets position 3 and third student gets position 1 and so on.

If $E$ = (3,2,1) then first student gets position 3rd, second student gets position 2nd, and third student gets position 1st, respectively.

Example 3:

An experiment is conducted to know the dosage of a medicine. The dosage is increased continuously until a patient reacts positively. One possible sample space for this experiment is to let $\Omega$ consist of all the positive numbers, so

$\Omega = (0,\infty)$

Where the outcome would be x if the patient starts getting the dosage and reacts when the value of dosage reaches x and no reaction to any smaller dosage than x.

The outcomes of a random experiment is called a simple event (or elementary event) and denoted by $\omega$ . The sample space $\Omega$ = $\{\omega_1, \omega_2, ... \omega_k \}$ is the set of all possible outcomes. And the subsets of $\Omega$ are called events and are denoted by capital letters, in general, such as $A, B, C$ .

Complementary Event

Suppose $\Omega_A$ contains all simple events that contain in the event A, then the event $A^c$ refers to the non-occurring of A and it is called a composite or complementary event.

Sure Event

If we consider in general, all the outcomes of a sample space can also be a part of an event. Since it contains all possible outcomes, we say that $\Omega$ will always occur and it is called a sure or certain event.

Impossible Event

If we consider the null set $\emptyset = \{\}$ as an event, then this event can never occur and it is called an impossible event.

The sure event therefore is the set of all elementary events, and the impossible event is the set with no elementary events.

Example 1:

Rolling a die, if a die is rolled once, then the possible outcomes are the number of dots on the upper surface of the die, that is 1,2,3,4,5,6.

Sample space is the set of simple events and these are $\omega_1$ = “1”, $\omega_2$ = “2”, $\omega_3$ = “3”, $\omega_4$ = “4”, $\omega_5$ = “5”, $\omega_6$ = “6”. Thus $\Omega$ = { $\omega_1, \omega_2, \omega_3,\omega_4,\omega_5,\omega_6$ }

Event A is an event of getting even number of dots on the upper surface of the die. There are three possibilities that this event occurs: $\omega_2, \omega_4,\omega_6$ .

Complementary Event of A: $\Omega_A$ it contains all the odd numbers. There are three possiblities that this event occurs.

Elementary Event is an event defined to observe only one particular number, say $\omega_1$ = “1”, then it is an elementary event.

Sure Event is the event that a number which is greater than or equal to 1 because any number between 1 and 6 is greater than or equal to 1.

Impossible Event is the event that the number is 7.

Example 2:

Rolling two dice: Suppose we throw two dice simultaneously. And an event is defined as the number of dots observed on the upper surface of both the dice. In this case, there are possibly 36 simple events and it is represented as $\omega_1$ = (1,1), $\omega_2$ = (1,2), …. $\omega_{36}$ = (6,6).

Suppose an event A is defined as upper faces of both the dice contain the same number of dots, then the sample space is $\Omega_A$ = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

Suppose an event B is defined as the sum of numbers on the upper faces is 6, then the sample space is $\Omega_B$ = {(1,5), (2,4), (3,3), (4,2), (5,1)}.

Q&A

From this note, we can get answers to the following questions.

What is a random experiment in probability?
How to construct an experiment, sample space and event in probability theory?
What are the different types of events in probability theory?

References

Essentials of Data Science With R Software – 1: Probability and Statistical Inference, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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Essentials of Data Science – Probability and Statistical Inference – Sample Space and Events

Experiment (or a random experiment)