Terms used in Probability: Definition, Terms, Examples

Probability is a topic of mathematics concerned with numerical representations of the chance of an event occurring or the truth of a statement. It is important to read each problem to define and interpret the events carefully. The first and most important step in understanding probability problems is to understand the terminology. If necessary, we need to reread the problem several times, clearly to identify the important event. In this article, we learn in detail about the terms used in probability.

Probability

The likelihood of an event occurring is defined by probability. We may be required to predict the outcome of an event in a variety of real-life situations. The outcome of an event may be certain or uncertain to us. In these instances, we state that the event has a chance of happening or not happening. In general, probability has many uses in games and business, where it may be used to produce probability-based predictions.

Probability Definition

The term “probability” refers to the measurement of an event’s likelihood. Because many events are impossible to anticipate with absolute certainty, probability can be used to estimate the possibility of an event occurring.

The theoretical probability (also called classical probability) of an event \(E\), written as \(P(E)\), is defined as

\(P(E)=\frac{\text { Number of favourable outcomes }}{\text { Total number of all possible outcomes }}\)

The probability value always lies between \(0\) and \(1\).

\(0 \leq P(E) \leq 1\)

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Terms Related to Probability

Experiment:

An experiment is a type of action with unknown outcomes. There are a few positive outcomes and a few negative consequences in every experiment.

Scientists will make thousands of unsuccessful attempts before they could make a successful attempt to make any invention.

Random Experiment:

A random experiment is one in which the set of possible outcomes is known. Still, the specific outcome in a given experiment cannot be predicted before the experiment is carried out.

Example: Rolling a die, tossing a coin

Trial: 

Trials are the various tries made during an experiment. In other words, a trial is any particular outcome of a random experiment.

 Example: Tossing a coin

Event:

A trial with a clearly defined outcome is an event. For example, getting a tail when tossing a coin is termed an event.

Random Event: 

A random event cannot be easily foreseen. The chance value for such situations is extremely low. The appearance of a rainbow in the rain is a completely random occurrence.

Outcome: 

The outcome of an event is a collection of all possible outcomes. 

Example: There are two different results when a sportsperson hits a ball towards the goal post. He has a chance to score or miss the goal.

Possible Outcome: 

The list of all possible outcomes of an experiment is known as the list of possible outcomes.

Example: Heads or tails are the two possible outcomes when tossing a coin.

Equally likely Outcomes:

Equally likely outcomes are the results of an experiment in which each of the possible possibilities has the same probability.

Example: When we roll a six-sided die, we have an equal chance of getting any number from \(1\) to \(6\).

\(\text {Probability of any number} =\frac{1}{6}\).

Sample Space:

The set of outcomes from all the trials in an experiment is known as the sample space.

The possible outcomes of rolling dice are \(1,2,3,4,5\), and \(6\).

The sample space is made up of these results.

\(~S~ = \left\{{1,~2,~3,~4,~5,~6}\right\}\)

Probable event: 

The term “probable event” refers to an event that can be predicted. We can calculate the probability of such happening.

Example: We can predict the likelihood of a child being promoted to the next class, calling this a probable event.

Impossible Event:

An impossible event is an event that is not a part of the experiment or does not happen in the sample space of the experiment’s outcomes.

For example, getting the number \(7\) on the face that is turned up is an impossible event when a die is thrown.

Complementary Events:

Complementary events happen when there are only two possible outcomes, one of them opposite the other.

For an event with the probability of \(P(E)\), its complement is \(P(\overline{E)}\).

\(P(E)+P(\overline{E)}=1\)

In general, it is true that for an event \(E\),

\(\Rightarrow P(\overline{E)}=1-P(E)\)

Or

\(\Rightarrow P(E)=1-P(\overline{E)}\)

The event \((\overline{E)}\), representing ‘not \(E\)’ is called the complement of the event \(E\). We also say that \(E\) and \((\overline{E)}\) are complementary events.

Mutually Exclusive Events:

Mutually exclusive events are those in which the happening of one event prevents the happening of the other. In other words, if two events cannot occur simultaneously, they are said to be mutually exclusive.

Example: Tossing a coin can result in either heads or tails. It is impossible to view both at the same time.

Solved Examples – Terms Used in Probability

Q.1. A dice is thrown \(70\) times, and \(4\) appeared \(21\) times. Now, in a random throw of a dice, what is the probability of getting a \(4\)?
Ans:
The total number of trials \(=70\).
The number of times \(4\) appeared \(=21\)
We know, the probability of an event\(=\frac{\text { Number of a favourable outcome }}{\text { Total number of the possible outcome }}\)
\(\Rightarrow \text {Probability of getting} \,4=\frac{\text { Number of times } 4 \text { appeared }}{\text { Total number of trials }}\)
\(\Rightarrow \text {Probability of getting} \,4=\frac{21}{70}=\frac{3}{10}\)
Therefore, \(\frac{3}{10}\) is the probability of getting a \(4\).

Q.2. If \(P(E)=0.06\), what is the probability of ‘not \(E\)‘ ?
Ans: We have \(P(E)=0.06, P(\overline{E)}=?\)
We know that for an event with the probability of \(P(E)\), its complement is \(P(\overline{E)}\).
We know \(P(E)+P(\overline{E)}=1\)
\(\Rightarrow P(\overline{E)}=1-P(E)\)
\(\Rightarrow P(\overline{E)}=1-0.06\)
\(\Rightarrow P(\overline{E)}=0.94\)
Therefore, \(0.94\) is the probability of ‘not \(E\)’.

Q.3. A football team plays \(120\) matches and wins \(70\) matches. What is the probability of the team winning the next match?
Ans:
Total number of matches played \(=120\)
Number of matches won by the team \(=70\)
We know that \(P(E)=\frac{\text { Number of outcomes }}{\text { Total number of all possible outcomes }}\)
\(\Rightarrow P(E)=\frac{70}{120}\)
\(\Rightarrow P(E)=\frac{7}{12}\)
Therefore, \(\frac{7}{12}\) is the probability of the team winning the next match.

Q.4. Find the experimental probability in a throw of dice of obtaining an odd number.
Ans: There are six possible outcomes in a die, which are \(1,2,3,4,5,6\).
Odd numbers on a die are \(1,3,5\)
\(\text {Probability of an event} =\frac{\text { Number of times an event occurs }}{\text { Total outcomes of trials }}\)
In this case, the probability of obtaining an odd number, \(P=\frac{3}{6}=\frac{1}{2}\)
Hence, the probability of getting an odd number on dice is \(\frac{1}{2}\).

Q.5. A coin is tossed \(1000\) times with the following frequencies: head: \(480\), tail: \(520\). Compute the probability of getting head.
Ans: The total number of trials is \(1000\) because the coin is tossed \(1000\) times.
The number of times the head occurs is \(480\).
\(\text {Probability of an event} =\frac{\text { Number of times an event occurs }}{\text { Total outcomes of trials }}\)
So, \(\text {probability(head)}=\frac{480}{1000}=\frac{12}{25}\)
\(\Rightarrow \text {probability(head)}=\frac{12}{25}\)
Therefore, the probability of getting head is \(\frac{12}{25}\).

Q.6. Madhu had a jar containing \(9\) red balls, \(5\) blue balls, and \(6\) green balls. She called one of her friends and asked them to pick a ball from the jar. What is the probability that the ball which is picked is either red or blue?
Ans: Let us take a count of the number of balls in the jar.
Number of red balls \(=9\)
Number of blue balls \(=5\)
Number of green balls \(=6\)
Total number of balls \(=20\)
Total number of red and blue balls \(=9+5=14\)
We know that \(P(E)=\frac{\text { Number of Outcomes }}{\text { Total mumber of all possible Out comes }}\)
In this case, the probability of getting a red or a blue ball is \(P(E)=\frac{14}{20}=\frac{7}{10}\)
Therefore, \(\frac{7}{10}\) is the probability that the ball which is picked is either red or blue.

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Summary

The set of outcomes from an experiment is known as an event in probability. Different terms are used in probability. Knowing about these terms is very important to understand probability and to solve different problems. This article includes the different terms used in probability and definition, examples of these terms.

This article, Terms used in Probability, might have helped in understanding these in detail, and it helps solve the problems based on these very easily.

Frequently Asked Questions(FAQs) – Terms Used in Probability

Q.1. What is the probability in simple terms?
Ans: A probability is a number that expresses the chance or likelihood of an event occurring. Probabilities can be stated as values ranging from \(0\) to \(1\) and percentages ranging from \(0\%\) to \(100\%\).

Q.2. What is an event in probability? Give an example.
Ans: The set of outcomes from an experiment is known as an event in probability. 
Example: Suppose we are experimenting with a coin flip.
The coin landing ‘heads’ or ‘tails’ is the result of this experiment. These are the events that occurred during the experiment.

Q.3. What is an impossible event in probability?
Ans: An impossible event cannot occur.
If and only if \(P(E)=0, \mathrm{E}\) is an impossible event.
Example: Getting both a head and a tail on a single coin flip is an impossibility.

Q.4. What is the outcome of the event?
Ans: The outcome of an event is a collection of all possible outcomes.
Example: Rolling a die to see if we get the desired number is appearing.

Q.5. What is a favourable outcome in probability?
Ans: The outcome of a random experiment is a possible outcome. A favourable outcome indicates the outcome that we expect to happen.

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