Explore probability theory across single events, two independent events, multi-trial series, algebraic solvers, and continuous standard normal distributions.
• Two Events: Enter P(A) and P(B) to evaluate intersection, union, symmetric difference, and conditional probabilities.
• Flexible Solver: Enter any 2 known values to automatically solve all 8 related probabilities.
• Series of Events: Add multiple event rows and repetition counts to calculate joint and "at least one" probabilities.
• Normal Distribution: Specify mean, standard deviation, and interval bounds (supports -inf and inf) with shaded bell curve visualizer.
To find out the union, intersection, and other related probabilities of two independent events.
Axioms of Kolmogorov, independent vs. dependent events, set algebra, and Gaussian normal distribution properties.
Probability is a mathematical framework for quantifying uncertainty. Formally established by Andrey Kolmogorov, all classical probability systems obey three fundamental axioms:
The probability that event A does NOT happen is one minus the probability that it does.
Two events are independent if the occurrence of one gives zero information about the likelihood of the other.
Subtracting the intersection prevents double-counting the overlapping region in the sample space.
The probability that either A occurs OR B occurs, but strictly NOT both simultaneously.
The normal (Gaussian) distribution is characterized by its symmetric bell curve centered at mean μ with spread determined by standard deviation σ. According to the Empirical Rule:
Mutually exclusive events cannot happen at the same time (P(A ∩ B) = 0). Independent events CAN happen together, but the occurrence of one does not change the probability of the other (P(A ∩ B) = P(A) × P(B)). If two non-zero events are mutually exclusive, they CANNOT be independent.
The easiest method is using the complement rule: P(at least one success) = 1 - P(no successes at all). For example, if a coin is flipped 5 times, P(at least 1 head) = 1 - (0.5)⁵ = 1 - 0.03125 = 0.96875 (96.88%).
A Z-score measures how many standard deviations an observation X is away from the mean μ: Z = (X - μ) / σ. A positive Z-score indicates a value above the mean, while a negative Z-score indicates a value below the mean.
No. By definition and mathematical axiom, all probabilities must fall strictly within the continuous range [0, 1] (or 0% to 100%). Values outside this range are invalid.