Introduction to Probability

Probability is a branch of mathematics that deals with numerical explanations of the chances of something happening or the accuracy of a statement. In general, the probability of an event is a number between 0 and 1, with 0 signifying impossibility and 1 indicating certainty. The greater the probability of something happening, the more likely it will happen.

The word probability comes from the Latin word probabilitas, which can imply “probity,” a measure of a witness’s authority in a court proceeding in Europe that is commonly linked to the aristocracy. In some ways, this differs significantly from the present definition of probability, which is a measurement of the weight of empirical evidence derived through inductive approach and statistical inference.

Probability of Events

The ratio of the number of favourable outcomes to the total number of outcomes of an event is known as a probability in Maths. The number of favourable outcomes can be expressed as x in an experiment with the ‘n’ number of outcomes.

Types of Probability

In terms of finding the probability of an event occurring, there can be different perspectives or types of probabilities based on the nature of the outcome or the method followed. They are as follows:

• Classical Probability
• Experimental Probability
• Subjective Probability
• Axiomatic Probability

1) Classical Probability

Using classical probability, known as “priority” or “theoretical probability,” an experiment with B equally likely outcomes results in an event X with exactly A of them, meaning the probability of X is: A / B. In other words, when a fair dice is rolled, six possibilities are equally likely to occur. This means that each number on the die has a 1/6 chance of coming up.

The formula for Classical probability is – 

P(A) = f / N is the mathematical form of the formula.

P(A) stands for “probability of event A,” 

 “f” denotes the frequency or how many times the event could occur.

The letter N represents the number of times the event could occur.

2) Experimental Probability 

This is based on observations made during an experiment. Divide the number of possible outcomes by the number of trials to calculate the experimental probability. For example, if a coin is tossed ten times and the head is recorded six times then, the experimental probability for heads is 6/10 or 3/5. The results of the probability calculations determine these probabilities. In scientific experiments, the probability is a function of the number of trials performed compared with the number of outcomes. The theoretical probability may differ from the actual value of probability based on experimental probability results.

The formula to calculate the experimental probability is: 

P(E) = Number of times an event occurs/Total number of times the experiment is conducted

3) Subjective Probability 

Individuals’ subjective probabilities are based on their beliefs about an event occurring. For example, the probability of a football team winning a match depends more on the fan’s beliefs and feelings than on formal calculations.

Subjective probability is based on opinions rather than calculations. They are mostly based only on past experiences.

For example, You’re taking your pet to the vet today, and based on past experience you’re sure the fees will exceed Rs. 500.

4) Axiomatic Probability

As part of axiomatic probability, we apply a set of rules or axioms to all types of events by Kolmogorov. Based on the applications of these axioms, one can quantify the likelihood of any event occurring or not occurring, as follows:

• Probability is the least possible at zero, and when it is at one, the probability is the highest
• Probability equals one for a certain event
• Only one of the events can occur in the union of events, while any two mutually exclusive events cannot co-occur

What is Conditional Probability?

A measure of the chance of an event occurring given that the other event (by presumption, assumption, statement, or evidence) has already happened is known as conditional probability.

Given two events A and B, conditional probability means the possibility of event B occurrence if event A has already occurred. Given that event A has happened, the conditional probability of event B occurring is:

 P(B/A) = P(A ∩ B)/P(A).

Examples of Probability in Real life

1. Weather Forecast – You utilise probability almost every day to organise your day around the weather. Meteorologists can’t predict the weather accurately; therefore, they rely on tools and instruments to anticipate rain, snow, or hail.
2. Sports Authority – Coaches and athletes use probability to determine the optimal game and competition strategy. A sports organisation, too, considers the likelihood of weather and other considerations while planning an event.
3. Insurance – When examining insurance policies and practises to ensure which programs are best for you and your family and what deductible levels you require, probability plays a significant part.
4. Games– You use probability when you play a board, card, or computer game that incorporates luck or chance. In order to receive the cards you want or the secret weaponry you need in a game, you must calculate the odds. What chance you’re willing to accept will be determined by the possibility of receiving those cards or tokens.

Conclusion

Check out these key probability points that summarise the key learnings on this topic.

• Probability is the degree to which a specific event is likely to occur
• Probability is always expressed as a fraction between 0 and 1
• The event can be viewed as a subset of the sample space
• When you throw a coin, you get heads or tails, and when you throw dice, you get 1, 2, 3, 4, 5, or 6
• The exact results of a random experiment cannot be predicted; only some probable outcomes can be predicted

In everyday life, the notion of probability is extremely important. This critical concept is the foundation of statistical analysis. Probability serves as a replacement for certainty in modern science.