What is Probability? — Section 1: Foundations of Probability

Probability is the mathematical framework we use to quantify, model, and reason about uncertainty. In many areas of life such as sports betting, weather forecasting, medical diagnosis, and financial markets we encounter situations with unknown or unpredictable outcomes.

Rather than making definitive claims, we express beliefs about outcomes using probabilities, which range from

0 \le P(A) \le 1

Here, $P(A)$ is the probability of event $A$ , where 0 means impossible and 1 means certain.

This allows us to make informed decisions by evaluating the likelihood of outcomes. For example, saying Team A has a 75% chance of winning corresponds to

P(\text{Team A wins}) = 0.75

This reflects strong expectation, while acknowledging uncertainty. The remaining uncertainty is

P(\text{Team A loses}) = 1 - 0.75 = 0.25

Probability helps incorporate randomness, variation, and incomplete information into models which is crucial when decisions must be made under uncertainty.

Rather than making definitive claims, we express beliefs about outcomes using probabilities, which range from

0 \le P(A) \le 1

Here, $P(A)$ is the probability of event $A$ , where 0 means impossible and 1 means certain.

This allows us to make informed decisions by evaluating the likelihood of outcomes. For example, saying Team A has a 75% chance of winning corresponds to

P(\text{Team A wins}) = 0.75

This reflects strong expectation, while acknowledging uncertainty. The remaining uncertainty is

P(\text{Team A loses}) = 1 - 0.75 = 0.25

Probability helps incorporate randomness, variation, and incomplete information into models which is crucial when decisions must be made under uncertainty.