Introduction to Probability

Definition

Probability is a numerical measure between 0 and 1 that represents how likely an event is to happen:

Values closer to 0 = more unlikely
Values closer to 1 = more likely
Formally, the probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times.

For equally likely events, probability can be calculated as:

Probability = \frac{Number of ways an event can happen}{Number of all possible outcomes}

Types of Probability

Subjective Probability
- Based on personal belief, experience, or feelings
- Different people may assign different probabilities
- Example: Probability of a sports team winning their next game
Theoretical Probability
- Based on formulas or assumptions about the event
- Often assumes events are equally likely
- Example: Probability of getting heads when flipping a fair coin = 1/2
Empirical Probability
- Based on collected data and observations
- Example: Flipping a coin 1000 times and finding the proportion of heads

Law of Large Numbers

As the number of trials increases, the empirical probability will converge to the true probability. This principle explains why casino games are profitable in the long run despite occasional player wins.