Probability Calculator
What is Probability?
Probability is a measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
The higher the probability of an event, the more likely it is to occur. Probability theory is used in statistics, mathematics, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Basic Probability Formulas
Simple Probability
P(A) = Number of favorable outcomes / Total number of possible outcomes
Example: The probability of drawing a king from a standard deck of cards is 4/52 = 1/13 ≈ 0.077 or 7.7%.
Probability of Multiple Events
P(A and B) = P(A) × P(B|A)
For independent events: P(A and B) = P(A) × P(B)
Example: The probability of flipping two heads in a row with a fair coin is 1/2 × 1/2 = 1/4 or 25%.
Probability of Either Event
P(A or B) = P(A) + P(B) - P(A and B)
For mutually exclusive events: P(A or B) = P(A) + P(B)
Example: The probability of drawing either a heart or a king from a standard deck is 13/52 + 4/52 - 1/52 = 16/52 = 4/13 ≈ 0.308 or 30.8%.
Complementary Probability
P(not A) = 1 - P(A)
Example: The probability of not rolling a 6 on a die is 1 - 1/6 = 5/6 ≈ 0.833 or 83.3%.
Types of Probability
Theoretical Probability
Based on the possible outcomes of an event. It assumes all outcomes are equally likely and is calculated using mathematical formulas.
Experimental Probability
Based on observations and experiments. It is calculated by dividing the number of times an event occurs by the total number of trials.
Subjective Probability
Based on personal judgment, experience, or belief about the likelihood of an event occurring.
Conditional Probability
The probability of an event occurring given that another event has already occurred.
Common Probability Distributions
Binomial Distribution
Used for situations with a fixed number of independent trials, each with the same probability of success.
Example: Flipping a coin 10 times and counting the number of heads.
Normal Distribution
A continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent than data far from the mean.
Example: Heights of people, IQ scores, measurement errors.
Poisson Distribution
Used for counting the number of times an event occurs in a fixed interval of time or space.
Example: Number of calls received by a call center per hour, number of defects in a manufactured product.
Applications of Probability
- Statistics and Data Analysis: Inferring population parameters from sample data.
- Finance and Insurance: Risk assessment, option pricing, insurance premiums.
- Medicine: Clinical trials, diagnostic testing, epidemiology.
- Weather Forecasting: Predicting the likelihood of various weather conditions.
- Games and Gambling: Calculating odds, expected values, and fair payouts.
- Machine Learning: Probabilistic models, Bayesian inference, uncertainty quantification.