Programs

# Subjective Probability: Function, Applications & Examples

## Introduction

Subjective probability is a type of probability wherein a specific outcome is likely to happen based on your judgment or experience. It helps you predict the outcome of an event either by referencing things that you have learned so far or based on your own experience.

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It is the exact opposite of objective probability, which measures the history of gathered data or recorded observations to predict if an event is going to occur or not. Subjective probability does not make use of mathematical calculation or data analysis; it rather depends upon your gut feeling to predict the outcome.

Subjective probability outcome will never be the same for two people as each person may have a different opinion or thinking about a particular event. For example, if two people were asked to predict how you would react in a specific situation, you will certainly hear different answers.

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Since the person closer to you knows you better, he/she would give a response based on your nature. The other person may consider different factors while predicting your response.

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## How Does the Subjective Probability Work?

Subjective probability has a high degree of personal bias, and the outcome of the probability is different for each person. In general, the probability is obtained by collecting the quantitative information or data and then interpreting the output by using some mathematical calculation or formula, which usually gives you a more accurate answer.

For example, you can predict the output of flipping a coin. There are always 50%-50% chances that the coin will land with a head and tail up.

On the other hand, the subjective probability is highly flexible and may not give you the correct answer as it is highly dependent on personal judgments held by each person. Judgments are based on an individual’s own experience and views. It often differs from person to person as they are subjective and based on how they interpret the situation given to him/her.

Though it does not use any statistical analysis or mathematical calculation, you can illustrate subjective probability as follows:

Probability (x) = degree of personal belief that x is true

Where x is an event, situation, or condition

## Uses of Subjective Probability

You might be wondering where we use subjective probability when it is so flexible and has no logical reasoning. Well, some industries find it useful and use these predictions to drive their business goals.

The subjective probability approach is used in multiple industries for decision-taking, such as marketing, economics, business, etc. For example, a sales manager predicts that there are 70% chances of getting the order for which his/her company has bid. Repeated tests or calculations cannot evaluate this percentage.

Some real-life examples where we use subjective probability are:

• Job interviews outcome
• Employee promotion
• Performance incentives

Subjective probability is highly affected by an individual belief or judgment about the likelihood of an event. The following are the disadvantages of subjective probability:

• The predictions are not backed up by logical reasoning or statistical calculation; it is always based on a high degree of bias.
• Subjective probability fails to meet the complex calculations.
• The outcome is never the same for an event or situation. For example, two or more individuals given the same situation may arrive at different outcomes, i.e., there may be different factors considered by the individuals for the same event.
• It must follow a few conditions to be workable. For example, when you predict the percentage of an event, whether it will occur or not, it must sum up to 100%.

Must Read: Types of Regression Models

## Examples of Subjective Probability

The following examples clearly state how subjective probability outcome differs for each person.

### Example 1

You are a huge fan of Virat Kohli, and the world cup cricket series is about to begin in a few days. You have been asked to predict the chances of India winning the World cup series. While there is no mathematical calculation or data to back up your predictions, you will still vouch for your favourite player or team in the actual percentage. For example, there is a 90% chance that India will win the World cup series.

### Example 2

You have been asked to predict the outcome of a flipping coin, whether it will land with head or tail up. Though, the mathematical calculation says that there is a 50% chance that it will land head up and a 50% chance that it will land tails up. In subjective probability, the percentage of your prediction may change based on the previous flips.

If the same coin has been flipped 15 times in the past and has given ten times heads up and five times tails up. You will say that there is a 75% chance of landing heads up. Though it is mathematically incorrect, your experience has created a situation that compels you to predict using subjective probability.

### Example 3

The weather department is predicting that it will rain in the next 2 hours based on wind pattern, weather situations, and their software analysis. But you may have the same predictions of rain in the next 2 hours based on your experience with weather or rain.

### Example 4

You have fallen sick and want to visit your family doctor. You want to predict how much money you should take while visiting a doctor. The last time when you visited him, the doctor was charging 500 Rs as a consultation fee.

But one of your family members informs you that he has upgraded his office to add facility, due to which his consultation charges might have gone up. You are now left with two options; either go by your budget or go without the idea of what the cost is going to be and end up spending more money. You will go by your gut instinct and choose the first option.

## Other Types of Probabilities

After looking at the subjective probability examples, you should also know that there are three other types of probabilities apart from subjective probability. They include:

Classical Probability

Other than objective and subjective probability, there is a classical probability, also known as theoretical probability. This probability is based on the theoretical chances of something happening. It believes x outcomes are likely to happen, and any event (y) has exactly z of these outcomes.

So, the probability of these outcomes happening in the event is z/x, which can also be stated as P(y) = z/x

It is also the first theory of probability that students learn in school. A common example is the chances of getting a number while rolling a normal die. There are six possible outcomes when rolling a fair die, out of which there is an equal possibility of getting one of the six results. Thus, when rolling a die, we have a 1/6 probability of rolling every number.

## Advantages and Drawbacks of Classical Probability

You can use classical probability when there is a finite number of outcomes in a situation, and it is also easy to calculate when compared to other types of probability.

The one drawback of theoretical probability is that you will not encounter many situations with a finite number of outcomes. For example, a rolling die has a finite number of outcomes, but something as abstract as the future has an infinite number of outcomes, so calculating the probability of future events becomes tough with classical probability.

Empirical Probability

Empirical probability is also known as experimental probability, as it uses thought experiments to define probability. Let’s understand this with an example:

You want to determine the probability of flipping heads on a coin. To do so, you experiment by flipping the coin 100 times and recording the results.

After flipping the coin, you observe that the coin landed on heads 56 times out of the 100 flips. To calculate the empirical probability of flipping heads, you divide the number of successful outcomes (56) by the total number of trials (100):

Empirical Probability of Flipping Heads = Number of Heads / Total Number of Flips

= 56 / 100

= 0.56

Therefore, based on the empirical data collected from the experiment, the probability of flipping heads is estimated to be 0.56 or 56%.

Thus, the empirical probability is the number of favourable outcomes divided by the total number of trials.

Axiomatic Probability

The axiomatic approach to probability provides a comprehensive framework that establishes a connection between theoretical and experimental probability, ultimately offering proof for subjective probability. This perspective encompasses a set of rules known as Kolmogorov’s three axioms, which apply to all probability perspectives. By employing axiomatic probability, it becomes possible to quantify the likelihood of an event taking place or not.

The three axioms can be applied to all other types of probability. Axiomatic probability is defined as the probability of any function when observed from number to events, which should then satisfy these three axioms.

1. One is the largest possible probability, and zero is the smallest.
2. A certain event has a probability of 1.
3. Two mutually exclusive events cannot take place at the same time, but the union of events implies that only one of them can happen.

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## Conclusion

To sum it all up, subjective probability is a type of probability based on individual knowledge, understanding, and experience of the likelihood of an event.  These predictions might be true if they are biased free and come up with some logical reasoning. But, there are situations, as explained throughout this article, that demands judgments or experience rather than calculations.

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## What is subjective probability?

It refers to how likely a particular event would happen, based on how an individual person values that event. This is different from objective probability, which is the proportion of all possible outcomes that a particular event will occur. For example, an individual has a small chance of winning a lottery, but he may view this outcome as very likely to happen, while another person may think that it is highly unlikely to win the lottery. The former attitude is subjective (or personal) probability, while the latter is objective probability.

## What are the applications of subjective probability?

Subjective probability follows the same rules as objective probability. The only differences are in how the probabilities are assigned. Subjective probability is used in decision making under uncertainty when the outcomes cannot be accurately predicted. Mathematics has been developed to formalize the concept of subjective probabilities. Probability theory is the mathematical basis of the theory of decision making under uncertainty. The Bayesian method uses subjective probabilities in order to make decisions. Subjective probability is used in social sciences such as psychology, sociology, political science, economics etc. It is also used in decision making like medical diagnosis, making business decisions etc.

## What is the difference between cumulative and subjective probabilities?

There are two types of probability. Cumulative probability and Subjective probability. Cumulative probability is the probability of an event over a series of trials. Subjective probability is your degree of belief in the event. The sum of subjective probabilities equals 100% because you are 100% sure that it will happen. It is just that you don't know when. There is always a chance of failure. So, in other words, you can't be 100% sure that you will make a million within 5 years. It is just 50/50 that you will make it.

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