What Everyone Should Know Should About Probability Definitions and Properties

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Most of us use this term on a daily basis. But how many know, exactly why we are using it? Probability is a branch of mathematics that concerns calculating the likelihood of any occurrence. The same is expressed in the range of 1 and 0.

For example, if the answer or result is 1, then the occurrence is considered a certainty. Any occurrence, which has a probability of 0, is not possible.

There are various kinds of definitions; however the main thing is the result. There are other definitions of probability as well.

‘The number of occurrences of a targeted event divided by the number of occurrences plus the number of failures of occurrences’.

There are several definitions, as well as interpretations of probability. This branch of mathematics, include discrete and continuous random variables, distributions etc.          

How did probability come into being?

The roots are set in analyzing game chance. This happened in the 16th century. Initially, the theory dealt with discrete events and its methods was a combination. At first, continuous variables was part of the theory.

Let Us Check Out the Properties of Probability

  • Firstly, when we set out to learn something new, there are a set of objectives, we have in mind.
  • We need to know, why learning probability is so important. How it is useful in scientific research.
  • You will learn the definition of any given event.
  • You will learn to find out new events, by considering subsets, unions, intersections and complements.
  • Learn about empty events, exclusive events, exhaustive events and more.
  • There are three approaches to probability. You will learn about those. The different approaches are person opinion approach, relative frequency approach, the classical approach.
  • You will learn about fundamental theorems and their applications.

The Properties

You need to keep in mind the basic properties, when working with probability theories. Let us get into the same:

  • 0 < P (A) < 1 – States a probability can never be larger than 1 or smaller than 0.
  • In case, events A and B, are mutually exclusive, we have that P (A+B+…) = P (A) +P(B)+..
  • Next, if the events occurring that is, A, B, are exclusive to one another, then we will have P( A+B+…) = P(A)+P(B)+… = 1.
  • Fourth instance is, if event A and B are independent of each other statistically. Then the result is – P(AB) = P(A) P (B) where P (AB) is called joint probability.
  • Fifth instance is, if A and B are not mutually exclusive, then P (A+B) = P (A) + P (B) – P(AB).


There are certain definitions that need to be taken care of. They are:

  • Experiment – it is a process, through which we can test different probabilities. Flip a coin; see what side comes up – heads or tails and how many times. Then drawing a card from another set of cards, the probability can be studied from here. The result is never certain, what might come up.
  • Outcome – It is the possible result of your experiment. Sometimes, it is positive and some times, it’s negative. The outcomes can never really be predicted.
  • Event – Here the different kinds of results are events. You get ahead, it is an event. You get a tail that is another event. It goes on like this.
  • Sample space – It is the total of all outcomes.
  • Equally Likely Events– if the two probabilities of two events are equal, then the two events that occur are considered equal.
  • Mutually Exclusive Events–If two events are not dependent on one another, or the happenings are independent of the other event, it’s called Mutually Exclusive Events.

If you thought, this was all, there is more.

  • Conditional Probability Properties is another important part of probability definitions and theory.
  • Mass Function is there. It is mainly concerned with defining discrete probability distribution. It is denoted by pmf.
  • Density Functionis connected to grossly random variables.

Now coming to the various approaches:

The personal opinion approach deals with things like, what you feel are likely. For e.g.

You think it will rain tomorrow, depending on the weather conditions and projections. You can even assign a percentage to that. Say 80% chances of rainfall is there. It is your personal approach to the issue at hand.

Then we all make assumptions regarding which will be the winning team, in any given match. Some say there is 1% chance, some say 90 % chance, and so on.

Then coming to Relative Frequency Approach, is second important approach. It involves performing an experiment with same set of parameters a number of times.

Count the number of times, different set of results occur.Then, weigh them against one another, analyses the parameters, when the same occurs. For example, toss a coin n number of times. Calculate the results – how many times heads come up and how many times, tails come up.

Next comes the classical approach. The whole scenario happens in classic scene.That means, in any given statistical event, the elements may be there, that have a certain probability of occurrence. The best example will be that of dice.Then, you roll a dice, 1, 2,3,4,5 or 6, any of the sides can come up. You can denote the probability as a fraction, in such cases. In this approach, you can divide the number of favorable outcomes by total number of possible outcomes and come to the result, that the fraction states.

Now, some people study probability as a hobby and some for necessity and education. It depends on person-specific qualities. Being in the mathematical realm, it can turn into a monster of some sort.

So, it is totally up to you, how you view probability. You either love it or hate it, there is nothing in between.

Now, if you want to chance games, you might as well study probability. It will help you determine,the chances of winning or losing. I you want to gamble, you need to study this subject. So, it’s up to you how you look at it!

A Detailed Study of the Definition of Probability and Its Properties

Probability is one of the topics that find a wide range of applications from high-school mathematics to high-end calculations to determine the future of some big multi-national company. On a personal, I found many of my peers to avoid this particular in our curriculum. The common misconception many people have regarding probability was it is tough to fathom; though the reality is something else.

The use of probability by the statistician to determine the growth and prospect of the company has been used for ages across the globe. People may find it surprising that using some simple calculations the statistician can help a company to predict its success in the future.

In this blog, we will try to breakdown every aspect and small details about probability so that we can sow a seed of interest in our reader’s mind and they can learn more about this unique calculation method.

What is probability?

The first question that may come to our mind is what is a probability?  The answer to the question is straightforward as by using this method we predict the likelihood of a particular event. Now, a possibility can never be less than zero and more than one. Both students and statisticians use the specific technique of verifying the calculation.

While using probability to calculate the first step a statistician makes is list down all the events that may occur during the particular time frame. The most common example teachers used to provide a better understanding of his disciple is the use of dice and all its possible outcomes. Suppose we are rolling two dices, there may be several results or being precise thirty-six outcomes. The thirty-six findings are known as the events, and it is the simple step of using this calculation. Once a student or statisticians master himself in listing out all the probabilities, the next few steps are just a cake walk.

The theorems of Cox and Kolmogorov:

On a higher level, there are several theories of probability and the most notable among them is Kolmogorov’s theory and Cox theory. In Kolmogorov’s opinion, the sets should get interpreted as events and probability will explain the class of those sets. In Cox’s view, he believed constructing a consistent set of probability values to proposition instead of one as he treated a single probability to be the first outcome. Both Kolmogorov and Cox followed similar law of probability, but on the grounds of technicality, they were poles apart.

Once we are successful in determining the possible outcomes of an event, then we categorize them into two columns; the first column lists down all the mutually exclusive events and the second column notes down all the Exhaustive outcomes. In some cases, a few findings fall under both the mentioned categories and they are called sample space.

The properties of probability:

It is easy to define probability by three properties but to help our readers we will incorporate a few examples so that it becomes easier for them to understand. We will use the most common case of a tossing die. Now, if we get asked what the probability of obtaining 1 to 6 face-up is. The sample space for the problem will be quite simple. It will look like {1, 2, 3, 4, 5, 6}. Now all these six outcomes are a sample point, but each is also mutually exclusive since only one number can be face up when we toss dice once. The same sample space is also exhaustive in nature as at least one of the six numbers will end up face-up. We will use E to denote the exhaustive outcomes, and they are E = {1, 3, 6}. We will now use F to donate the possibility of getting 6 as the outcome, and it will look like F = {6}. The space of event which we used to denote F is a set of the subset, or in other words, each element of F is an event. The three properties of probability are:

  • Range:

This property helps us to determine the range of all the possibilities. For Example in the case of rolling dice, the range of possibility for the event will range from 1 to 6.

  • Sure thing:

This property tells us that a probability can never be more than 1. Now if we roll a dice, the chance of getting a sample is 1 out of possible 6; it can never be 7 out of possible 6.

  • Sigma- additivity or countable additivity:

In case of mutually exclusive events, the addition or union of all the events will give us the probability measure denoted by P.

The sigma- additivity is a bit complicated process and requires constant attention to get hold of it, so any student finds our blog interesting may surely hit the term on Google and learn more about it.

In this section of our blog, we will try to mention some of the old definitions of probability used though none of them are as accurate as the above mentioned as none of them were able to clarify the meaning of probability.

  • The classical definition of probability:

The classical definition stated that if the chance of getting all the possible outcomes is equally likely, the likelihood of that particular event is calculated using the ratio of a specific result and the total number of results.

  • Frequentist definition of probability:

The Frequentist definition stated that the possibility of getting an outcome depends on the frequency of the result occurring in that period when observed over a large number of repetitions.

  • Subjective definition of probability:

The description stated that the possibility of getting an event depends on the willingness of a person to take a chance on that event. The biggest flaw of this definition was it lacked objectivity as the desire of making a bet for a particular event varies differently for each person.

Author Bio

Nancie L Beckettis a former professor and has an MBA degree, along with 6 years of work experience. She graduated from Duke University and has experience of teaching high school students as well as college level students. She also has a degree in mathematics that makes her an expert in probability theory.

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