1.1 Probability and The Real World
The inherent complexity and unpredictability of natural and human systems cause uncertainty. Many phenomena are influenced by numerous variables, interactions, and external factors that can give rise to unpredictable outcomes. Additionally, our knowledge and understanding of the world are limited, and there may be gaps in our information or hidden factors that we are unaware of, further contributing to uncertainty.
In the words of Benjamin Franklin - “In this world, there is nothing certain but death and taxes”.
Let's go through a few examples to understand how probability helps us make better decisions in the face of uncertainty in the real world.
Consider a GPS app that provides two paths between points and . Can we be certain about which path is the shortest? Similarly, consider an insurance company that provides term insurance coverage for 30 years.
Can we be certain that a death benefit will be paid to the beneficiary if the policyholder passes away during the coverage period while the policy is active?
In these examples, do we need probability to find the shortest path or determine if the beneficiary will receive the death benefit? The shortest path can be calculated through mathematical tools, while the beneficiary is guaranteed to receive the death benefit if the policyholder dies within the coverage period as stated in the policy's terms.
However, let's look at these examples from a different perspective. In the same GPS app, can the company be certain about the fastest path? Similarly, can the insurance company be certain that the person will not die within the coverage period and that the company will not have to pay the insured amount?
Do you sense some uncertainty here? Many unknowns come into play. Traffic conditions can change dynamically, and factors such as accidents, road closures, congestion, or unexpected events can impact the travel time on both paths. Likewise, a person's age can be affected by various factors like eating habits, exercise, environment, genes, and various other reasons.
Despite the uncertainty in these situations, the GPS app can still provide a reasonably accurate idea of the fastest route, and the insurance company can devise different plans and remain profitable. How do they achieve this? What is happening here?
It appears that these companies are using probability to make efficient predictions. In its simplest form, the GPS app takes into account historical data, peak traffic hours, congestion, accidents, past passenger travel times, and various other factors affecting travel time. By integrating this data with real-time traffic information, the system creates a probabilistic model indicating how likely a path is to be the fastest. These predictions are inherently probabilistic and subject to uncertainties, but they still provide a good estimate of the fastest route. Similarly, the insurance company has considered factors like age, eating habits, lifestyle, medical conditions, and various other elements to create probabilistic models that can estimate how likely is it for the policyholder to pass away within the term of the insurance policy. Based on this estimation, the company determines the premium to charge its policyholders.
Based on what we have seen so far, what can we say about probability? Can we define probability as:
“Probability is how likely something is to occur”
The definition "Probability is how likely something is to occur" may seem like an intuitive way to describe probability but lacks a concrete and quantitative basis for measuring how likely something is to occur. Therefore, akin to many other measurable attributes, like temperature, distance, time, weight etc., describing probability in numerical terms provides a standardized and objective approach to measuring how likely something is to occur. So, our probability definition updates to:
“Probability is a numerical description of how likely something is to occur”
|
In probability space, the term "something" mentioned in the earlier statement is precisely defined as an "Event." The section on Event Space (1.4.2) will provide a comprehensive understanding of this concept. |
Probability is represented by a numerical value within the range of 0 and 1 or, expressed as a percentage between 0% and 100%. As the probability of something happening increases, so does its numerical value. A completely impossible event has a probability of 0, whereas an event that is certain to occur has a probability of 1.
|
In section 1.5, we’ll see why the probability lies between the range of 0 and 1. |
When we are dealing with probability, there are different ways in which we express it. Consider a statement “There is a 40% probability of rain tomorrow.” Alternatively, the same statement can also be expressed as:
- "There is a 40% chance of rain tomorrow."
- "It is 40% likely to rain tomorrow."
|
Expressing the statement in the following ways is incorrect and misrepresents probability:
|
Some more statements illustrating how we use probability:
- “The chances of winning the lottery jackpot are incredibly slim, maybe around 1 in 300 million”.
- “The probability of passing the UK driving test on the first attempt is approximately 52.6%.”
- "The risk of being involved in a car accident increases in rainy weather, so I'll drive more cautiously."
- "The risk of contracting a certain disease is estimated to be 10% for the general population."
- "The chances of getting a job offer after the interview are about 1 in 10."
|
"Probability" and "chance" are synonymous and used interchangeably in the real world |
Probability is a fundamental concept that permeates different areas of our daily lives. However, the broad spectrum of probability has led to its diverse interpretations. In the upcoming section, we’ll explore the various ways in which we arrive at probability and interpret it.