Probability Calculator
Calculate two independent events, solve any 2 known probabilities, analyze series of events, and find normal distribution interval areas with dynamic SVG curves.
Probability Calculator
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.
Probability of Two Events
To find out the union, intersection, and other related probabilities of two independent events.
Comprehensive Guide to Probability Theory & Distributions
Axioms of Kolmogorov, independent vs. dependent events, set algebra, and Gaussian normal distribution properties.
1Foundations of Probability & Kolmogorov Axioms
Probability is a mathematical framework for quantifying uncertainty. Formally established by Andrey Kolmogorov, all classical probability systems obey three fundamental axioms:
- Non-negativity: For any event E, 0 ≤ P(E) ≤ 1.
- Unit Measure: The probability of the entire sample space S is P(S) = 1.
- Countable Additivity: For any countable sequence of mutually exclusive events, the probability of their union is the sum of their individual probabilities.
2Core Set Theory Formulas & Probability Rules
Complement Rule
The probability that event A does NOT happen is one minus the probability that it does.
Multiplication Rule (Independent)
Two events are independent if the occurrence of one gives zero information about the likelihood of the other.
General Addition Rule
Subtracting the intersection prevents double-counting the overlapping region in the sample space.
Symmetric Difference
The probability that either A occurs OR B occurs, but strictly NOT both simultaneously.
3Normal Distribution & Empirical 68-95-99.7 Rule
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:
- 68.27% of data falls within 1 standard deviation (μ ± 1σ).
- 95.45% of data falls within 2 standard deviations (μ ± 2σ).
- 99.73% of data falls within 3 standard deviations (μ ± 3σ).
Frequently Asked Questions
What is the difference between independent and mutually exclusive events?
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.
How do you calculate the probability of "at least one" success in repeated trials?
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%).
What does a Z-score represent in a normal distribution?
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.
Can a probability be greater than 1 or negative?
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.
Understanding Probability Calculator & Formula
The Probability Calculator on CalculatorScale.com is engineered to deliver instant, mathematically verified calculations conforming strictly to standard institutional equations. Whether planning long-term investments, managing household budgets, tracking clinical fitness targets, analyzing academic genetics, calculating veterinary dosages, or converting dimensional units, every calculation updates dynamically in real time without lag.
Every formula used in this tool adheres to certified international standards, including World Health Organization biometric metrics, veterinary clinical guidelines, NIST physical conversion constants, and standard biological equations. Results are computed entirely client-side inside your browser with complete data privacy, ensuring your sensitive numbers and statistics never leave your device.
Our calculators are 100% free with no registration, subscriptions, or paywalls required. You can toggle between imperial and metric units, customize input parameters, export calculation summaries, or embed this calculator directly into your own website or blog.
How to Use the Probability Calculator
- 1 Choose a module: "Two Events", "Two Events (Flexible Solver)", "Series of Events", or "Normal Distribution".
- 2 For Two Events: Enter P(A) and P(B) to evaluate intersection, union, symmetric difference, neither, and 2-circle Venn diagram.
- 3 For Flexible Solver: Provide any 2 known probability values to solve the remaining 6 fields algebraically.
- 4 For Series of Events: Add rows with individual event probabilities and repetition counts (k) to compute joint occurrence.
- 5 For Normal Distribution: Enter mean (μ), standard deviation (σ), and left/right bounds (accepts -inf/inf) to view shaded bell curve.
Probability Calculator — Frequently Asked Questions
Q1 What is the difference between independent and mutually exclusive events? ▾
Mutually exclusive events cannot occur simultaneously (P(A ∩ B) = 0). Independent events can occur together, and the occurrence of one does not alter the likelihood of the other (P(A ∩ B) = P(A) × P(B)).
Q2 How do you calculate the probability of "at least one" event happening in multiple trials? ▾
Use the complement rule: P(at least one) = 1 - P(none occur). For independent events with probability p over k trials: P(at least one) = 1 - (1 - p)^k.
Q3 How does the normal distribution calculate area under the curve? ▾
The area under the normal curve represents cumulative probability and is calculated using the error function: Area = Φ(Z₂) - Φ(Z₁), where Z = (X - μ) / σ.
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