Probability
Statistics & ProbabilityProbability 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; 0.5 means equally likely to happen or not.
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 $1/6$.
Key Insight
Probability answers the question "How likely is it?" A higher probability means it is more likely to happen.
Definition
Probability is a measure of the likelihood of an event, expressed as a number between $0$ and $1$. For equally likely outcomes: $P(\text{event}) = \text{favorable outcomes}/\text{total outcomes}$. Probability rules: $0 \le P(A) \le 1$, $P(\text{certain event}) = 1$, $P(\text{impossible event}) = 0$.
Example
From a standard deck of $52$ cards, the probability of drawing a heart is $13/52 = 1/4 = 0.25$. The probability of drawing a face card (Jack, Queen, King) is $12/52 = 3/13$.
Key Insight
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 $0.7+0.3=1$.
Definition
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 $P(\bigcup A_i) = \sum P(A_i)$. All properties of probability follow from these three axioms.
Example
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. This formalization allows probability theory to handle both discrete and continuous sample spaces rigorously.
Key Insight
The frequentist interpretation views $P(A)$ as the long-run relative frequency of $A$ in infinitely many trials. The Bayesian interpretation views $P(A)$ as a subjective degree of belief, updated via Bayes' theorem as new evidence arrives. Both interpretations satisfy Kolmogorov's axioms.