1.2.3 Subjective Probability

Subjective probability is a way to express how likely we think something is, even if we do not have concrete data or statistical evidence to support our beliefs. It is based on an individual's subjective perceptions, opinions, and experiences, rather than on objective data or statistical analysis. This type of probability is influenced by a person's knowledge, past experiences, emotions, and intuition. It represents the degree of confidence or uncertainty an individual assigns to an outcome, often based on their understanding of the situation at hand.

Example 1.7. Let's analyze the "coin-flipping" experiment from the perspective of subjective probability. Imagine you're holding a standard, fair coin in your hand, and you're about to flip it. Your subjective probability of the coin landing heads or tails is influenced by your personal beliefs, experiences, and perceptions. Here are a few factors that might shape your subjective probability in this scenario:

• Past Experiences: Your past experiences with coin flips may influence your subjective probability. For instance, if you've flipped a coin many times before and noticed that it tends to land on heads more often, you might have a subjective bias toward heads.
• Intuition: Your gut feeling or intuition can also play a role. You might have a hunch or feeling that the coin is more likely to land on one side based on factors like how you flipped it or the surface it's landing on.
• Psychological Factors: Your mood, emotions, or psychological state at the moment of the flip can impact your subjective probability. If you're feeling particularly lucky or unlucky, it might affect your perception of the outcome.
• Personal Beliefs: Your personal beliefs or superstitions can come into play. For example, if you have a lucky charm associated with one side of the coin, you might believe that it's more likely to land that way.
• External Factors: External factors, such as someone commenting on the coin flip or the context in which it's happening (e.g., a decision-making situation), can also influence your subjective probability.

Example 1.8. For instance, suppose your company develops an innovative artificial intelligence-based product and you have no idea if your customers will like it or not. In this case, you may rely on your intuition or previous experiences and learnings from other products launched earlier by your company, even if they are not directly comparable to this new product. By analyzing the marketing strategy, customer acquisition approach, and any other relevant information, you draw inferences and form a view that there is a high probability (say 90%) that the new product will be successful. Alternatively, your colleague may assign a much lower probability (say 40%) to the product being successful based on her belief that it is not cost-effective or the timing of the product is not ideal, among other factors.

Since this probability is subjective, one person's probability may differ from another's. This can be unsettling for many people. However, it reflects the reality that people often differ in their judgments of probability.

In our exploration of probability, we have discussed the three fundamental definitions: theoretical probability, empirical probability, and subjective probability. These definitions help us comprehend how we quantify uncertainty and assess the probability of random processes based on different approaches—mathematical models, observed data, and personal beliefs, respectively.

It's now time to explore another fundamental concept that forms the backbone of probability: Sets, which is covered in the next section