Probability Calculator

Two events are independent if the occurrence of one does not affect the probability of the other (e.g., flipping two separate coins; P(A ∩ B) = P(A) × P(B)). Mutually exclusive (disjoint) events cannot happen simultaneously (e.g., rolling a 2 and a 5 on a single die roll; P(A ∩ B) = 0). In this calculator, toggling the relationship switch dynamically updates the intersection and union formulas accordingly.

This is a classic misunderstanding known as the Gambler's Fallacy. Each attempt is an independent trial. After 100 attempts with a 1% (0.01) chance, the probability of failing every single time is (1 - 0.01)¹⁰⁰ = (0.99)¹⁰⁰ ≈ 36.6%. Consequently, your cumulative chance of obtaining at least one drop is 1 - 0.366 = 63.4%—significantly lower than certainty.

Expected Value is the long-term average outcome of an event if repeated many times. It is calculated as EV = (P(Win) × Gain) - (P(Loss) × Cost). A positive EV (green) indicates an advantageous bet or investment where you profit on average over the long run, whereas a negative EV (rose) indicates an inevitable financial drain over repeated trials.

Yes. The input parser automatically detects forward slashes (e.g., '1/6' for rolling a specific die face, or '1/52' for drawing a specific card) as well as percentages (e.g., '25%') and raw decimals (e.g., '0.05'). The calculator simplifies and converts the value dynamically.

The intersection P(A ∩ B) represents the probability that both Event A and Event B occur together. For independent events, it equals P(A) × P(B). The union P(A ∪ B) represents the probability that at least one of the events occurs, given by the Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For mutually exclusive events, the intersection is 0, so P(A ∪ B) = P(A) + P(B).

To find the number of attempts n needed for a target probability P, use the formula n = ln(1 - P) / ln(1 - p), where p is the single-trial drop rate. For example, with a 1% drop rate (p = 0.01), achieving 95% confidence requires n = ln(0.05) / ln(0.99) ≈ 299 attempts, and 99% confidence requires approximately 459 attempts.

No. According to Kolmogorov's Axioms of Probability, the probability of any event must strictly lie within the interval [0, 1] (or 0% to 100%). A probability of 0 denotes an impossible event, while 1 denotes absolute certainty. The calculator automatically clamps all inputs and outputs within this mathematically valid range.

The PDF export generates an official branded report including a header with generation timestamp, the active analysis mode, all configured input values (drop rate, attempt counts, event probabilities, reward/loss sums), and the final calculated probabilities, color-coded for quick verification and presentation.