Probability
Probability is a number between 0 and 1 that measures the likelihood that a specific event will occur.
Formula
\text{P(event)} = \dfrac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}}
Definition
Probability is a number from 0 to 1 that describes how likely something is to happen: a probability of 0 means impossible, 1 means certain, and 0.5 means equally likely to happen or not. For equally likely outcomes: $P(\text{event}) = \text{favorable outcomes}/\text{total outcomes}$, following the rules $0 \le P(A) \le 1$, $P(\text{certain event}) = 1$, and $P(\text{impossible event}) = 0$. Formally, in Kolmogorov's axiomatic framework, a probability space is a triple $(\Omega, \mathcal{F}, P)$ where $\Omega$ is the sample space, $\mathcal{F}$ is a sigma-algebra of events, and $P: \mathcal{F} \to [0,1]$ is a probability measure satisfying $P(\Omega)=1$, $P(\emptyset)=0$, and countable additivity $P(\bigcup A_i) = \sum P(A_i)$, from which all properties of probability follow.
Example
When you flip a fair coin, the probability of heads is $1/2 = 0.5 = 50\%$; when you roll a fair die, the probability of getting a $4$ is $1/6$. From a standard deck of $52$ cards, the probability of drawing a heart is $13/52 = 1/4 = 0.25$, and the probability of drawing a face card is $12/52 = 3/13$. The Borel sigma-algebra on $\mathbb{R}$ is the smallest sigma-algebra containing all open intervals, and Lebesgue measure restricted to $[0,1]$ gives the uniform probability measure, formalizing probability theory for both discrete and continuous sample spaces.
Key Insight
Probability answers the question "How likely is it?" and all probabilities for all possible outcomes must add up to $1$; if $P(\text{rain}) = 0.7$, then $P(\text{no rain}) = 0.3$, because $0.7+0.3=1$. The frequentist interpretation views $P(A)$ as the long-run relative frequency of $A$ in infinitely many trials, while the Bayesian interpretation views it as a subjective degree of belief updated via Bayes' theorem as new evidence arrives; both interpretations satisfy Kolmogorov's axioms.