Probability Toolkit

Two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if and only if P(A ∩ B) = P(A) × P(B). In other words, the joint probability equals the product of individual probabilities. Equivalently, P(A|B) = P(A), meaning knowing B occurred doesn't change the probability of A. Examples of independent events include: flipping two coins (outcome of first flip doesn't affect second), rolling two dice, or drawing cards with replacement. Non-independent (dependent) events include: drawing cards without replacement, weather on consecutive days, or the probability of traffic accidents given icy road conditions.

Combinations (nCr) count selections where order does NOT matter, while permutations (nPr) count arrangements where order DOES matter. For example, choosing 3 people from 5 for a committee uses combinations (5C3 = 10) because the order of selection doesn't matter—{Alice, Bob, Carol} is the same committee as {Carol, Alice, Bob}. However, assigning 3 people from 5 to President, Vice President, and Secretary positions uses permutations (5P3 = 60) because order matters—Alice as President with Bob as VP is different from Bob as President with Alice as VP. Formula difference: nCr = n!/(r!(n-r)!) divides by r! to eliminate order, while nPr = n!/(n-r)! keeps all orderings. Generally, nPr ≥ nCr with equality only when r=1.

Choose based on your scenario: (1) Binomial Distribution: Use when you have a fixed number of independent trials (n), each with constant success probability (p), and you're counting successes. Examples: coin flips, quality control with fixed sample size, survey yes/no responses. (2) Poisson Distribution: Use for counting rare events occurring in a fixed interval of time/space, with events happening independently at a constant average rate (λ). Examples: customer arrivals per hour, network failures per day, typos per page. (3) Geometric Distribution: Use when counting trials until the first success in repeated independent trials. Examples: number of product tests until first defect, lottery tickets until first win, sales calls until first conversion. (4) Hypergeometric Distribution: Use when sampling without replacement from a finite population with two categories. Examples: drawing cards without replacement, quality control sampling without replacement, polling without replacement. Key distinction: Binomial uses replacement (constant p), Hypergeometric doesn't (changing probabilities).

Large factorial, combinatorial, or power calculations produce extremely large numbers that exceed typical decimal display limits. For example, 100! ≈ 9.33 × 10^157 (a number with 158 digits), and 1000C500 is astronomically large. Scientific notation (e.g., 1.23e+45 meaning 1.23 × 10^45) provides a compact, readable format that prevents overflow errors and maintains precision. This notation consists of a mantissa (significant digits) and an exponent (power of 10). To interpret: 5.2e+8 = 520,000,000 and 3.1e-4 = 0.00031. You can switch to scientific notation in the display settings when working with large n values (typically n > 100) to ensure all results display correctly. This is standard practice in statistical software and scientific computing.

Small discrepancies in cumulative probability calculations typically result from floating-point precision limitations inherent in computer arithmetic. Computers represent numbers with finite precision (typically 64-bit doubles with ~15-17 significant digits), which can accumulate small rounding errors through repeated addition or subtraction operations. For example, CDF values are computed by summing PMF probabilities: P(X ≤ k) = P(X=0) + P(X=1) + ... + P(X=k). Each addition can introduce a tiny error (~10^-15), and these errors can accumulate slightly. You might see P(X ≤ 100) = 0.999999999999998 instead of exactly 1.0. These differences are negligible for practical purposes (typically < 10^-12) and don't affect the validity of statistical conclusions. If you need exact arithmetic, symbolic computation tools can be used, but for nearly all real-world applications, the precision provided is more than sufficient.

Bayes' Theorem updates prior probability P(H) of a hypothesis H into a posterior probability P(H|E) after observing evidence E. The formula is: P(H|E) = [P(E|H) × P(H)] / P(E), where P(E|H) is the likelihood of observing evidence E given hypothesis H is true, P(H) is the prior probability before seeing evidence, P(E) is the total probability of observing the evidence (normalizing constant), and P(H|E) is the posterior probability after incorporating evidence. Example: Medical testing. Prior: P(disease) = 0.01 (1% prevalence). Likelihood: P(positive test | disease) = 0.95 (sensitivity). We also need P(positive test | no disease) = 0.10 (false positive rate = 1 - specificity). Then P(positive test) = P(positive | disease)×P(disease) + P(positive | no disease)×P(no disease) = 0.95×0.01 + 0.10×0.99 = 0.1085. Finally, P(disease | positive test) = (0.95 × 0.01) / 0.1085 ≈ 0.0876 or 8.76%. Despite a positive test, the probability of having the disease is only ~9% due to the low base rate (1% prevalence). This counterintuitive result demonstrates the importance of considering base rates in diagnostic reasoning.

The Random Simulator performs Monte Carlo trials to approximate probabilities through repeated random sampling, demonstrating the Law of Large Numbers and providing empirical validation of theoretical results. It generates thousands of random outcomes based on your probability model and calculates the observed frequency of events. As the number of trials increases, the simulated probability converges to the theoretical value. For example, flipping a fair coin 10 times might give 6 heads (60%), but 10,000 flips will likely yield very close to 50% heads. This tool is valuable for: (1) Approximating complex probabilities where analytical formulas are intractable, (2) Validating theoretical calculations by comparing with empirical results, (3) Visualizing probability convergence and variability, (4) Teaching concepts like sampling distributions and the Law of Large Numbers, (5) Estimating probabilities in games, competitions, or scenarios with complex rules. The simulator shows both the convergence plot (how probability stabilizes with more trials) and histogram (distribution of outcomes), providing visual insight into random processes.