In this post, I will discuss the Basics of Probability theory. First I will start with the concept of Set and Event.
Table of Contents
Set
In statistical theory, a set is a well-defined collection of distinct events. For example, whenever a coin is tossed or die rolled, something (event) will happen. Distinct events comprise the set, that is when a coin is tossed, either Hear or Tail. It can be denoted with a Set.
$$A=\{Head, \, Tail\}$$
Similarly, for a fair die, the distinct events can be represented as set $B$, that is,
$$B = \{1, 2, 3, 4, 5, 6\}$$
When two fair dice are rolled, there will be 36 events that can be represented in a set say $C$.
\begin{align*}
\Big\{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\,\, &(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\,\, &(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
\,\, &(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\,\, &(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
\,\, &(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\Big\}
\end{align*}
Basics of Probability
Probability is the chance of occurrence of an event described in a set (or sample space). For example, what is the chance of rain today? what is the chance that Pakistan will win the T20 World Cup? Probability is the estimation of chance and it deals with the occurrence of an event in the future. The estimates are presented numerically. For example, (i) There is a 75% chance of rain today, (ii) The insurance industry requires precise knowledge about the risk of loss to calculate premiums, and (iii) The chances of winning the lottery game are 1 in 2.3 million.
Random Experiment
Regarding probability, it is important to understand the concept of random experiments. It is a planned process/activity that gives different results known as outcomes. For example, as discussed earlier, when a coin is tossed, there may be two possible outcomes, Head or Tail. Any experiment or planned process which has only one outcome cannot be regarded as a random experiment. A random experiment has at least a minimum of two outcomes. Outcomes are the results of the experiment. The random experiment has the following properties:
- It can be repeated any number of times practically or theoretically.
- Each experiment has a minimum of two possible outcomes.
- All the outcomes are known in advance but each outcome is unpredictable.
So, we can say that probability is the measure of the degree of uncertainty or quantification of uncertainty.
Sample Space
When we collect all possible outcomes, it is known as sample space, and represented by $S$. For example,
$S=\{H, T\}$ sample space for tossing a single coin
$S=\{HH, HT, TH, TT\}$ sample space when tossing two coins simultaneously
Each outcome of a sample space is called a sample point.
Event
The individual outcome from a sample space in which one is interested is called an event. Events may be based on a single sample point or more than one sample point. For example, Let the even be even numbers when a single dice is thrown, that is, $A=\{2, 4, 6\}$, or even maybe $T$ (Tail) when tossing a coin. $B=\{T\}$. When we throw two dice the event may be the same number on the upper face of the dice, $C=\{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\}$
Similarly, the sum of dots on the top face of two dice is 4 is another event, that is, $D=\{(2, 2), (1, 3), (3, 1)\}$
Types of Events
The probability of an event lies between 0 and 1 inclusive. If the probability of an event is 1, it is known as a sure event. If the probability of an event is zero it is an impossible event. When two or more events cannot occur at the same time it is called a mutually exclusive event. For example, in the coin tossing example, either $H$ will occur or Tail, both head and tail cannot occur at the same time.
Events are equally likely when events have the same chance of occurrence. For example, either a student will pass or fail, there is a 50% chance for both events. Collectively Exhaustive Events are events whose union is equal to the sample space.
Random Variable
A random variable is that which takes values randomly. A random variable may be represented by $X$, $Y$, and $Z$, etc. Random variables can be classified as discrete random variables or continuous random variables. A discrete random variable is based on a counting procedure, while a continuous random variable is based on measurements.
A random variable is a variable that takes values randomly. These values may be integers for discrete variables and real for continuous variables. When we toss a coin there may be $H$ or $T$. Suppose, you are interested in Head then the random variable may be denoted by $X$ for various numbers of heads (for example, 0 head and 1 head)
| x(Heads) | $P(X)$ |
|---|---|
| 0 head | $\frac{1}{2}$ |
| 1 head | $\frac{1}{2}$ |
| Total | $1.0$ |
The sample space is $S=\{H, T\}$.
For two coins
| x(Heads) | $P(X)$ |
|---|---|
| 0 head | $\frac{1}{4}$ |
| 1 head | $\frac{2}{4}$ |
| 2 heads | $\frac{1}{4}$ |
| Total | $1.0$ |
The sample space is $S=\{HH, Ht, TH, TT\}$.
