Terms in Probability

Probability is a part of mathematics that deals with the likelihood of different outcomes occurring. It plays an important role in estimating the outcome or predicting the chances of that event. It is useful in real-life applications that are useful in solving complex problems and provide insightful insights.

In this article, we'll learn some important elementary 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. It expresses how likely an event is to occur and takes the value between $0$ and $1$.

Probability (Event) = Favorable Outcomes / Total number of outcomes

$P(E)=\frac{n(E)}{n(S)}$

Random Experiment

An experiment is called a 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 coins are tossed simultaneously, then possible outcomes are

• Heads on both coins $= (H,H) = HH $
• Head on the first coin and Tail on the other $= (H,T) = HT $
• Tail on the first coin and Head on the other $= (T,H) = TH $
• Tail on both coins $= (T,T) = TT $

Thus, the sample space is $S = {HH, HT, TH, 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 the occurrence of prime numbers, $B$ is the event of the occurrence of odd numbers and $C$ is the event of the occurrence of even numbers.

Also, observe that $A, B,$ and $C$ are subsets of $S$.

Now, what is the occurrence of an event?

From the above example, the experiment of throwing a die. Let E denote 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 $\omega \in \mathrm{E}$. 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 $=A=\{H\}$
   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 heads $=\mathrm{A}=\{\mathrm{HH}\}$
   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:$ ‘At least one head appeared’

$C:$ ‘Atmost one head appeared’ etc.

are all compound events.

The subsets of $S$ associated with these events are

$ S = \{HHH, HHT, HTT, HTH, THH, THT, TTH, TTT\}$

$A = \{HHT, HTH, THH\}$

$B = \{HTT, THT, TTH, HHT, HTH, THH, HHH\}$

$C = \{TTT, THT, HTT, TTH\}$

Each of the above subsets contains 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 that 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 event $E$.

Thus, the event $E = φ$ is an impossible event.

Now let us take up another event F “The number that turns up is less than $7$”.

$F = \{1, 2, 3, 4, 5, 6\} = S$ i.e., all outcomes of the experiment ensure the occurrence of the event F. Thus, the event $F = S$ is sure.

Solved examples Based on Probability:

Example 1: Which of the following is NOT an experiment?

1) Tossing a coin

2) Selecting a good student from class

3) Selecting a card from $52$ cards

4) Selecting a color out of $V, I, B, G, Y, O, R$

Solution:

Experiment - An operation that results in some well-defined outcomes is called an experiment.

Since the term "good" is not well-defined, it is not an experiment.

Hence, the answer is the option (2).

Example 2: Which of the following is NOT a random experiment?

1) Toss a coin.

2) Roll a die.

3) Turn on the right.

4) Record the number of students in city.

Solution

The statement in option (3) suggests that a person needs to turn to the right, so the outcome of this experiment can be predicted in advance. So this is not a random experiment.

Hence, the answer is the option (3).

Example 3: Which of the following is NOT an event of the random experiment of rolling a die?

1) Getting a number divisible by $3$.

2) Getting a multiple of $7$.

3) Getting an even prime.

4) Getting an odd prime.

Solution

Sample Space:

$S=\{1,2,3,4,5,6\}$

Since it is not possible to get a multiple of $7$ on rolling a die, it is not an event of the random experiment of rolling a die.

Hence, the answer is the option (2).

Example 4: Which of the following is NOT an event?

1) Getting a prime number on die.

2) Getting two heads on a coin.

3) Getting an even number on a die.

4) Getting two jacks from a deck of cards.

Solution

Since we can not get two heads on a single coin toss, therefore it is not an event.

Hence, the answer is the option (2).

Eexample 5: Which of the following is NOT a simple event?

1) Event of team winning a match.

2) Event of choosing a card from $52$ cards.

3) Getting an even prime number on dice.

4) Getting an odd number on dice.

Solution

Since there are $3$ odd numbers on a die, so in this event we have:

$\{1, 3, 5\}$

Hence, this is not a simple event.

Hence, the answer is the option (4)

Summary

Probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes. 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. The set of outcomes from an experiment is known as an Event. By understanding important terminologies is a fundamental concept. These terms helps to solve complex problems in mathematics. These methods are widely used in real-life applications providing insights and solutions to complex problems. Mastery of these concepts can help in solving gaining deeper insights and contributing meaningfully to real-life problems.