Probability: A Brief Overview
Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is a numerical measure that ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
Key Concepts
- Experiment: A procedure that results in an observable outcome.
- Outcome: A possible result of an experiment.
- Sample Space: The set of all possible outcomes of an experiment.
- Event: A subset of the sample space.
- Probability of an Event:
The ratio of the number of favorable outcomes to the total number of possible outcomes.
Types of Probability
- Theoretical Probability: Based on theoretical considerations, such as the equal likelihood of outcomes in a fair coin toss.
- Experimental Probability: Based on observed data from repeated experiments.
Probability Rules
- Addition Rule: For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
- Multiplication Rule: For independent events A and B, P(A and B) = P(A) * P(B).
- Conditional Probability: The probability of event A given that event B has already occurred.
- Complement Rule: P(not A) = 1 - P(A).
Probability Distributions
- Discrete Probability Distribution: A table or formula that lists all possible values of a discrete random variable and their corresponding probabilities.
- Continuous Probability Distribution: A function that describes the probability of a continuous random variable taking on a value within a given interval.
Common Probability Distributions
- Binomial Distribution: Used for counting the number of successes in a fixed number of independent Bernoulli trials.
- Poisson Distribution:
Used for modeling the number of occurrences of an event in a fixed interval of time or space. - Normal Distribution: A bell-shaped curve used to model many real-world phenomena.
Example:
- Tossing a coin: The sample space is {heads, tails}.
- Probability of getting heads: 1/2.
- Probability of getting heads or tails: 1.