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Whenever we’re unsure about the outcome of an event, we talk about the probabilities of its outcome. You might have heard people saying ‘it will probably rain today’, ‘I will probably pass the board exams’, ‘there is a high probability of getting a storm tonight’, and ‘chances of my winning in coming tournaments are very dim’, and so on.
Let’s understand what is probability and how is it related to chance with examples.
What is Probability?
There are many events as seen above that cannot be predicted. Probability is a way of associating a number with a chance or likelihood of an event. It is a measure of the likelihood of an event occurring. Using the concept of probability we can predict the chance of an event to occur i.e., how likely it is going to happen. The probability of an event can range from $0$ to $1$, where $0$ means an event is an impossible event and $1$ means an event is a certain event. The values $0$ and $1$ can also be expressed as percentages ranging from $0 \% \left( = 0 \right)$ to $100 \% \left(= 1 \right)$.
Probability can simply be said to be the chance of something happening, or not happening. So the chance of an occurrence of a somewhat likely event is what we call probability.
Examples of Probability
Ex 1: If you roll a six-faced die, then the chance of rolling a six is $\frac{1}{6}$.
$0 \le \frac{1}{6} \le 1$ (Probability of an event lies between 0 and 1)
Ex 2: If you toss two coins, then the probability of getting one head and one tail is $\frac{1}{2}$.
$0 \le \frac{1}{2} \le 1$ (Probability of an event lies between 0 and 1)
Ex 3: In a deck of well-shuffled cards, the probability of getting a King is $\frac{1}{13}$.
$0 \le \frac{1}{13} \le 1$ (Probability of an event lies between 0 and 1)
Note:
- Probability of an event lies between $0$ and $1$ (both inclusive)
- Probability of an event cannot be a negative number
Probability Definition Math
After understanding the basic concept of probability, let’s look at the formal definitions of the term probability. As learned above probability is a measure of the likelihood of the occurrence of an event when a certain experiment is performed. Hence, probability can be defined as:
- Probability is a mathematical term for the likelihood that something will occur. It is the ability to understand and estimate the possibility of a different combination of outcomes.
- Probability means that it is possible. It is a branch of statistics that deals with the occurrence of a random event. The number is expressed from 0 to 1 where 0 represents an impossible event and 1 represents a sure event.
- Probability is basically the degree to which something can happen. In order to determine the probability of a single event occurring, first of all, we need to know the total number of possible consequences.
Basic Concepts of Probability
Let’s now understand how the probability of an event is calculated. But before that let’s understand some basic terminologies associated with probability.
Experiment
In statistics, an experiment is defined as an ordered procedure that is performed with the objective of verifying, and determining the validity of the hypothesis. Before performing any experiment, some specific questions for which the experiment is intended should be clearly identified.
Examples of Experiment
Ex 1: Picking a card from a deck of well-shuffled cards.
Ex 2: Adding 45 and 23 using a calculator.
Ex 3: Rolling a pair of dice.
Random Experiment
An experiment that has a well-defined set of outcomes and the exact result or outcome cannot be predicted beforehand is called a random experiment. It is also called a trial. For example, when we toss a coin, we know that we would get a head or a tail, but we are not sure which one will appear.
Examples of Random Experiment
Ex 1: Picking a card from a deck of well-shuffled cards.
Ex 2: Rolling a pair of dice.
Note: Adding 45 and 23 using a calculator is an experiment but not a random experiment because the result(or outcome) will always be 68, which is a fixed value and not a random value.
Outcome
The result of any random experiment is called an outcome. Suppose a coin is tossed once we get a head(H) as the upper surface. So, tossing a coin is a random experiment that results in an outcome ‘head’ (or ‘H’).
Examples of Outcome
Ex 1: If you roll a six-faxed die the possible outcomes are 1, 2, 3, 4, 5, and 6.
Ex 2: If you toss a pair of coins, the possible outcomes are (Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail).
Equally Likely Outcomes
The outcomes with the same theoretical probability (or likelihood) of occurring are referred to as equally likely outcomes. For example, when a coin is tossed once, the relative occurrences of Head and Tail are equal. So, Head and Tail are equally likely outcomes that make the tossing of a coin fair and unbiased if it’s to decide between two options.
Examples of Equally Likely Outcomes
Ex 1: When a coin is tossed once, the relative occurrences of Head and Tail are equal. So, Head and Tail are equally likely outcomes that make the tossing of a coin fair and unbiased if it’s to decide between two options.
Ex 2: When a six-faced die is rolled, the relative occurrences of 1, 2, 3, 4, 5, and 6 are equal. So, getting 1, 2, 3, 4, 5, and 6 are equally likely outcomes. Here the likelihood of getting 1, 2, 3, 4, 5, or 6 is $\frac{1}{6}$.
Sample Space
A sample space is a collection of all possible outcomes of an experiment. A sample space is represented in the form of a set in a roster form.
Examples of Sample Space
Ex 1: When a coin is tossed once, the sample space is S = {Head, Tail} or S = {H, T}.
Ex 2: When a six-faced die is rolled once, the space is S = {1, 2, 3, 4, 5, 6}.
Ex 3: When a pair of coins are tossed, or a single coin is tossed twice, the sample space S = {(Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail)} or {(HH), (HT), (TH), (TT)}.
Sample Point
Each of the single elements of a sample space is called a sample point of a sample space.
Examples of Sample Point
Ex 1: In a sample space when a coin is tossed once, ‘Head’(or ‘H’) is a sample point. Similarly, ‘Tail’(or ‘T’) is also a sample point.
Ex 2: In a sample space when a die is rolled once, the sample points are 1, 2, 3, 4, 5, and 6.
Random Variable
A random variable is a variable that can take on many values. This is because there can be several outcomes of a random occurrence. Thus, a random variable should not be confused with an algebraic variable. An algebraic variable represents the value of an unknown quantity in an algebraic equation that can be calculated. On the other hand, a random variable can have a set of values that could be the resulting outcome of a random experiment.
Examples of Random Variable
Ex 1: Two dice are rolled and the random variable, X, is used to represent the sum of the numbers. Then, the smallest value of X will be equal to 2 (1 + 1), while the highest value would be 12 (6 + 6). Thus, X could take on any value between 2 to 12 (inclusive). Now if probabilities are attached to each outcome then the probability distribution of X can be determined.
Event
Any subset of a sample space of a random experiment is called an event.
Examples of Event
Ex 1: When a pair of coins are tossed, the sample space S = {(Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail)} and for this sample space, there are many subsets such as {(Head, Head)}, {(Head, Tail)}, (Head, Tail), (Tail, Head)}, etc. All these subsets are called events when a pair of coins are tossed.
Ex 2: When a die is rolled once, the sample space is S = {1, 2, 3, 4, 5, 6} and for this sample space, there are many subsets such as {1}, {2}, {3}, {4}, {5}, {6}, {1, 2}, {1, 3}, {2, 5}, {3, 4}, {1, 2, 5}, {3, 4, 6}, …. All these subsets are called events of when a die is rolled.
Favourable Event
An outcome that makes necessary the happening of an event in a trial is called a favorable event.
Examples of Favourable Event
Ex 1: When a die is rolled, the favourable events of getting an odd number are {1}, {3}, or {5}.
Ex 2: When a pair of dice is rolled, the favourable events of getting a sum of 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
Unfavourable Event
An outcome that prohibits(or restricts) the happening of an event in a trial is called an unfavorable event.
Examples of Unfavourable Event
Ex 1: When a die is rolled, the unfavourable events of getting an odd number are {2}, {4}, or {6}.
Ex 2: When a pair of dice is rolled, the unfavourable events of getting a sum of 7 are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 6), (3, 1), (3, 2), (3, 3), (3, 5), (3, 6), (4, 1), (4, 2), (4, 4), (4, 5), (4, 6), (5, 1), (5, 3), (5, 4), (5, 5), (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), and (6, 6).
Note: When in a sample space
- the number of favourable events is equal to the number of unfavourable events, then the associated event is equally likely to happen.
- the number of favourable events is greater than the number of unfavourable events, then the associated event is very likely to happen.
- the number of favourable events is less than the number of unfavourable events, then the associated event is very unlikely to happen.
Exhaustive Events
Exhaustive events are a set of events in a sample space such that one of them compulsorily occurs while performing the experiment. In simple words, we can say that all the possible events in a sample space of an experiment constitute exhaustive events.
The events $\text{E}_1$, $\text{E}_2$, $\text{E}_3$, $\text{E}_4$, …, $\text{E}_n$ are called exhaustive events, if $\text{E}_1 \cup \text{E}_2 \cup \text{E}_3 \cup \text{E}_4 \cup … \text{E}_n = \text{S}$, where $\text{S}$ is a sample space.
Examples of Exhaustive Events
Ex 1: When a coin is tossed, there are two possible outcomes – head(H) or tail(T). So, these two outcomes are exhaustive events as one of them will definitely occur while flipping the coin. And also $\{\text{H}\} \cup \{\text{T}\} = \{\text{H}, \text{T}\}$ is a sample space.
Ex 2: When a die is tossed, then the events ‘getting an odd number’ and ‘getting an even number’ are exhaustive events as the outcome will always be an odd number or an even number. The event ‘getting an odd number’ is $\{\text{1, 3, 5}\}$ and the event ‘getting an even number’ is $\{\text{2, 4, 6}\}$ and $\{\text{1, 3, 5}\} \cup \{\text{2, 4, 6}\} = \{\text{1, 2, 3, 4, 5, 6}\}$ is a sample space.
Mutually Exclusive Events
The events that cannot happen simultaneously are called mutually exclusive events.
Examples of Mutually Exclusive Events
Ex 1: When in a random experiment of tossing a coin, the events ‘Head’(or ‘H’) and ‘Tail’(or ‘T’) are mutually exclusive events. A coin cannot land on both head and tail when it is tossed.
Ex 2: When a card is picked from a deck of playing cards, the events ‘getting a king’ and ‘getting a jack’ are mutually exclusive events, as a card cannot be a king and a jack.
Ex 3: When a card is picked from a deck of playing cards, the events ‘getting a king’ and ‘getting a black card are non-mutually exclusive events, as a card can be a king and black also. (Getting a black king, i.e., king of spade or king of club).
Ex 4: When a die is tossed then the events ‘getting an odd number’ and ‘getting a prime number’ are non-mutually exclusive events, as the numbers 3 and 5 are odd numbers as well as prime numbers.
Impossible Event
An event that cannot happen when a random experiment is performed is called an impossible event.
Examples of Impossible Event
Ex 1: Getting the number ‘7’ when a six-faced die is rolled is an impossible event. The probability of an impossible event is always 0.
Ex 2: When a pair of dice is rolled, then ‘getting a sum of 15’ is an impossible event, since, the maximum sum that one can get is 12(6 on the first die, 6 on the second die).
Certain Event
An event that definitely will happen when a random experiment is performed is called a certain event.
Examples of Certain Event
Ex 1: Getting a number greater than 0 and less than 7 when a six-faced die is rolled is a certain event. The probability of a certain event is always 1.
Ex 2: When a pair of dice is rolled, then ‘getting a sum between 2 and 12’ is a certain event, since, the minimum sum is 2(1 on the first die, 1 on the second die) and the maximum sum is 12(6 on the first die, 6 on the second die)
Note: The probability of an event always lies between 0 and 1 (both inclusive). If $\text{E}$ is an event, and the probability of the event $\text{E}$ is $\text{P}(\text{E})$, then $0 \le \text{P}(\text{E}) \le 1$.
Computing the Probability of an Event
Depending on the approach used to compute the probability there are different ways of computing the probability of an event. The four different approaches to computing the probability of an event are
- Classical Approach: The classical or theoretical approach to probability states that in an experiment where there are $n$ equally likely outcomes, and event $\text{E}$ has exactly $m$ of favourable outcomes, then the probability of $\text{E}$ is $P( \text{E}) = \frac{m}{n}$. This is often the first perspective that students experience in formal education. For example, when rolling a fair die, there are six possible outcomes that are equally likely, you can say there is a $\frac{1}{6}$ probability of rolling each number.
- Empirical Approach: The empirical or experimental approach to probability defines probability by actually performing the experiment. For example, if you want to find the probability of getting a ‘Head’ when a coin is tossed, you toss the coin a number of times (performing an actual experiment), then count the number of times the coin landed on ‘Head’. The formula used to compute the probability of an event $\text{E}$ is $P(\text{E}) = \frac{a}{b}$, where, $b$ is the number of times an experiment is performed(or a number of trials) and $a$ is the number of times the event has occurred.
- Subjective Approach: The subjective approach to probability considers a person’s own personal belief or judgment that an event will happen. For example, an investor may have an educated sense of the market and intuitively talk about the probability of a certain stock going up tomorrow. You can rationally understand how that subjective view agrees with theoretical or experimental views. In other words, it’s the probability that what a person is expecting to happen through their knowledge and feelings will actually be the outcome, with no formal calculations.
- Axiomatic Approach: The axiomatic approach to probability is a unifying perspective where the coherent conditions used in theoretical and experimental probability prove subjective probability. You apply a set of rules or axioms by Kolmogorov to all types of probability. There are Kolmogorov’s three axioms that are used. When using axiomatic probability, you can quantify the chances of an event occurring or not occurring.
Uses of Probability
Probability is important to figure out if a particular thing is going to occur in an event or not. It also helps us to predict future events and take action accordingly. Below are the uses of probability in our day-to-day life.
- Weather Forecasting: We often check weather forecasting before planning for an outing. The weather forecast tells us if the day will be cloudy, sunny, stormy, or rainy. On the basis of the prediction made, we plan our day.
- Agriculture: Temperature, season, and weather play an important role in agriculture and farming. Weather forecasting, helps the farmers to do their job well on the basis of predictions. Undoubtedly, the occurrence of erratic weather is beyond human control, but it is possible to prepare for adverse weather if it is forecasted beforehand.
- Politics: You might be aware of the term ‘Exit Polls’. Whenever elections happen and results are yet to be declared, we want to predict the outcome of an election even before the polling is done. These predictions are called Exit Polls and use the concept of probability.
- Insurance: Insurance companies use probability to find out the chances of a person’s death by studying the database of the person’s family history and personal habits like drinking and smoking. Probability also helps to examine and evaluate the best insurance plan for the benefit of a person and his family.
Practice Problems
- Define the following terms
- Probability
- Random Experiment
- Event
- Exhaustive Events
- Favourable Events
- Mutually Exhaustive Events
- State True or False
- The probability of an event lies between $0$ and $1$ (both inclusive).
- The probability of an event lies between $0 \%$ and $1 \%$ (both inclusive).
- The probability of an event lies between $0$ and $100$ (both inclusive).
- The probability of an event lies between $0 \%$ and $100 \%$ (both inclusive).
- The events that cannot happen simultaneously are known as exhaustive events.
- The events that cannot happen simultaneously are known as mutually exclusive events.
FAQs
What is the simple definition of probability?
A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%
What is probability with example?
Probability is a value associated with the chance of occurring an event. For example, the probability of flipping a coin and its being heads is $\frac{1}{2}$, because there is $1$ way of getting a head and the total number of possible outcomes is $2$ (a head or tail). We write P(heads) = $\frac{1}{2}$.
Who is the father of probability?
Blaise Pascal and Pierre de Fermat are regarded as fathers of probability as they laid the fundamental groundwork of probability theory.
What is the concept of probability?
Probability is a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
Who gave the concept of probability?
Blaise Pascal and Pierre de Fermat laid the fundamental groundwork of probability theory, and are thereby accredited as the fathers of probability.
How the concept of probability started?
The concept of probability developed in a very strange manner. In 1654, a gambler Chevalier de Mere approached the well-known 17th-century French philosopher and mathematician Blaise Pascal regarding certain dice problems. Pascal became interested in these problems, studied them, and discussed them with another French mathematician, Pierre de Fermat. Both Pascal and Fermat solved the problems independently. This work was the beginning of Probability Theory.
What are the 4 types of probability?
The four perspectives on probability commonly used are Classical, Empirical, Subjective, and Axiomatic.
a) Classical: The classical or theoretical approach to probability states that in an experiment where there are $n$ equally likely outcomes, and event $\text{E}$ has exactly $m$ of favourable outcomes, then the probability of $\text{E}$ is $\text{P} (\text{E}) = \frac{m}{n}$.
b) Empirical: The empirical or experimental approach to probability defines probability by actually performing the experiment. The formula used to compute the probability of an event $\text{E}$ is $\text{P} (\text{E}) = \frac{a}{b}$, where, $b$ is the number of times an experiment is performed(or a number of trials) and $a$ is the number of times the event has occurred.
c) Subjective: The subjective approach to probability considers a person’s own personal belief or judgment that an event will happen.
d) Axiomatic: The axiomatic approach to probability is a unifying perspective where the coherent conditions used in theoretical and experimental probability prove subjective probability.
What is the concept of probability distribution?
A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. This range is bounded between the minimum and maximum possible values, but precisely where the possible value is likely to be plotted on the probability distribution depends on a number of factors such as the distribution’s mean (average), standard deviation, skewness, and kurtosis.
How are probability and statistics related?
Probability deals with predicting the likelihood of future events, while statistics involves the analysis of the frequency of past events. Probability is primarily a theoretical branch of mathematics, which studies the consequences of mathematical definitions.
Where do we use the probability formula in real life?
The following are some of the applications of probability in real life:
a) Weather Forecasting
b) Agriculture
c) Politics
d) Insurance
Conclusion
Probability is a mathematical term for the likelihood that something will occur. It is the ability to understand and estimate the possibility of a different combination of outcomes. The probability of an event can range from $0$ to $1$, where $0$ means an event is an impossible event and $1$ means an event is a certain event. The four different approaches to computing probability are classical, empirical, subjective, and axiomatic.
