Probability Calculator

Combinations are selections where order doesn't matter (C(n,r) = n!/(r!(n-r)!)), while permutations are arrangements where order matters (P(n,r) = n!/(n-r)!). For example, choosing 3 people from 10 for a committee uses combinations, but arranging 3 people in specific positions uses permutations. The number of permutations is always greater than or equal to combinations for the same values.

Bayes theorem helps calculate the probability of having a disease given a positive test result, considering the test's accuracy and disease prevalence. Formula: P(Disease|Positive) = P(Positive|Disease) × P(Disease) / P(Positive). This accounts for false positives and gives more accurate diagnostic probabilities than just the test accuracy alone, especially important for rare diseases.

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. Formula: P(A|B) = P(A∩B)/P(B). It's used in weather forecasting, medical testing, quality control, machine learning, and any scenario where one event affects the likelihood of another. For example, the probability of rain given cloudy skies is higher than the general probability of rain.

Businesses use probability for risk assessment, investment analysis, insurance pricing, market forecasting, and decision-making under uncertainty. Examples include calculating default risks for loans, stock return probabilities, success rates for new product launches, and Value at Risk (VaR) calculations. Probability models help quantify uncertainty and make data-driven decisions in volatile markets.

Probability is used in weather forecasting (chance of rain), sports betting (odds calculation), insurance (risk assessment), medicine (diagnosis accuracy), quality control (defect rates), artificial intelligence (machine learning algorithms), traffic planning (route optimization), and everyday decisions. It helps quantify uncertainty and make informed choices in situations with unknown outcomes.

For independent events, multiply individual probabilities. Coin flip: P(heads) = 1/2. Two coins both heads: (1/2) × (1/2) = 1/4. Dice: P(6) = 1/6. Two dice both 6: (1/6) × (1/6) = 1/36. For multiple outcomes, add probabilities of each favorable outcome. For example, rolling a 4 or 5 on one die: P(4) + P(5) = 1/6 + 1/6 = 1/3.

Theoretical probability is calculated based on the mathematical analysis of an event, assuming perfect conditions. For example, the theoretical probability of flipping heads is 1/2. Experimental probability is based on actual trials and observations. If you flip a coin 100 times and get 47 heads, the experimental probability is 47/100. As the number of trials increases, experimental probability approaches theoretical probability.

False positive: Test is positive but condition is absent (Type I error). False negative: Test is negative but condition is present (Type II error). Sensitivity measures the ability to correctly identify positive cases, while specificity measures the ability to correctly identify negative cases. These concepts are crucial in medical testing, quality control, and statistical hypothesis testing to understand test accuracy and reliability.

The base rate fallacy occurs when people ignore the prior probability (base rate) of an event when updating beliefs with new information. For example, if a disease affects 1 in 10,000 people and a test is 99% accurate, a positive result doesn't mean 99% chance of having the disease. Using Bayes theorem, the actual probability might be much lower due to the low base rate. This fallacy is common in medical testing and legal reasoning.

For multiple independent events occurring together (AND), multiply their probabilities. For at least one event occurring (OR), use: P(A or B) = P(A) + P(B) - P(A and B). For independent events, P(A and B) = P(A) × P(B). To find the probability of at least one success in n trials: P(at least one) = 1 - P(all failures) = 1 - (1-p)^n, where p is the probability of success in each trial.

Prior probability is your initial belief about an event before considering new evidence. Posterior probability is the updated belief after incorporating new evidence using Bayes theorem. For example, in medical diagnosis, the prior might be the general disease prevalence (1%), and after a positive test, the posterior is the updated probability of having the disease (which depends on test accuracy). This framework allows for systematic belief updating with new information.

Probability calculations are accurate when the underlying assumptions are met: events are truly independent, probabilities remain constant, and all relevant factors are considered. Real-world accuracy depends on data quality, model assumptions, and environmental stability. While theoretical probabilities provide excellent frameworks for decision-making, practical applications may involve additional uncertainties, changing conditions, and unmeasured variables that can affect outcomes.