Important Terminologies and Definitions of Probability - Practice Questions & MCQ

First, we learn some important elementary technical terms related to probability and discuss some examples concerning these.

Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes.

$
\begin{aligned}
&\text { Probability }(\text { Event })=\text { Favorable Outcomes } / \text { Total number of outcomes }\\
&P(E)=\frac{n(E)}{n(S)}
\end{aligned}
$

Random Experiment

An experiment is called random experiment if it satisfies the following two conditions: 

1. It has more than one possible outcome. 

2. It is not possible to predict the outcome in advance.

An experiment whose all possible outcomes are known but the outcome in one experiment cannot be predicted with certainty.

For example, when a coin is tossed it may turn up a head or a tail (so we know the possible outcomes), but we are not sure which one of these results will actually be obtained.

Sample Space

A possible result of a random experiment is called its outcome and the set of all possible outcomes of a random experiment is called Sample Space. Generally, sample space is denoted by S.

Each element of the sample space is called a sample point. In other words, each outcome of the random experiment is also called a sample point.

1. Rolling of an unbiased die is a random experiment in which all the possible outcomes are $1,2,3,4,5$ and 6 . Hence, the sample space for this experiment is, $S=\{1,2,3,4,5,6\}$.
2. When two coin is tossed simultaneously, then possible outcomes are
- Heads on both coins $=(\mathrm{H}, \mathrm{H})=\mathrm{HH}$
- Head on first coin and Tail on the other $=(H, T)=H T$
- Tail on first coin and Head on the other $=(\mathrm{T}, \mathrm{H})=\mathrm{TH}$
- Tail on both coins $=(\mathrm{T}, \mathrm{T})=\mathrm{TT}$

Thus, the sample space is $S=\{\mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \mathrm{TT}\}$

Event

The set of outcomes from an experiment is known as an Event.

When a die is thrown, sample space $S=\{1,2,3,4,5,6\}$.
Let $A=\{2,3,5\}, B=\{1,3,5\}, C=\{2,4,6\}$
Here, A is the event of occurrence of prime numbers, B is the event of occurrence of odd numbers and C is the event of occurrence of even numbers.
Also, observe that $A, B$ and $C$ are subsets of $S$.
Now, what is Occurrence of an event?
From the above example, experiment of throwing a die. Let E denotes the event " a number less than 4 appears". If any of '1' or '2' or '3' had appeared on the die then we say that event E has occurred.
Thus, the event $E$ of a sample space $S$ is said to have occurred if the outcome $\omega$ of the experiment is such that $\omega \in E$. If the outcome $\omega$ is such that $\omega \notin E$, we say that the event $E$ has not occurred.

Mutually Exclusive Events
Two or more than two events are said to be mutually exclusive if the occurrence of one of the events excludes the occurrence of the other

Independent Events
Events can be said to be independent if the occurrence or non-occurrence of one event does not influence the occurrence or
non-occurrence of the other.

Simple Event

If an event has only one sample point of a sample space, it is called a simple (or elementary) event.

1.When a coin is tossed, sample space $S=\{H, T\}$

The event of an occurrence of a head $=\mathrm{A}=\{\mathrm{H}\}$
The event of an occurrence of a tail $=B=\{T\}$
Here, $A$ and $B$ are simple events.|
2. When a coin is tossed two times, sample space $S=\{\mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \mathrm{TT}\}$

The event of an occurrence of two head $=\mathrm{A}=\{\mathrm{HH}\}$
The event of an occurrence of two tail $=B=\{T T\}$
Here, $A$ and $B$ are simple events.

Compound Event

If an event has more than one sample point, it is called a Compound event.

For example, in the experiment of “tossing a coin thrice” the events

 A: ‘Exactly one tail appeared’ 

B: ‘Atleast one head appeared’ 

 C: ‘Atmost one head appeared’ etc.

are all compound events. 

The subsets of S associated with these events are

$S=\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTT}, \mathrm{HTH}, \mathrm{THH}, \mathrm{THT}, \mathrm{TTH}, \mathrm{TTT}\}$
$\mathrm{A}=\{\mathrm{HHT}, \mathrm{HTH}, \mathrm{THH}\}$
$B=\{H T T, T H T, T T H, H H T, H T H, T H H, H H H\}$
$\mathrm{C}=\{\mathrm{TTT}, \mathrm{THT}, \mathrm{HTT}, \mathrm{TTH}\}$

Each of the above subsets contain more than one sample point, hence they are all compound events

Impossible and Sure Events 

Consider the experiment of rolling a die. The associated sample space is

$
S=\{1,2,3,4,5,6\}
$
Let E be the event " the number appears on the die is greater than 7 ".
Clearly no outcome satisfies the condition given in the event, i.e., no element of the sample space ensures the occurrence of the event E .
Thus, the event $E=\varphi$ is an impossible event.
Now let us take up another event $F$ "the number that turns up is less than 7 ".
Clearly, $F=\{1,2,3,4,5,6\}=S$ i.e., all outcomes of the experiment ensure the occurrence of the event $F$. Thus, the event $F=\mathrm{S}$ is a sure event.