SamCalculator

Probability Calculator

Calculate event probabilities, unions, intersections, complements, independent event outcomes, normal distribution probabilities, confidence intervals, and step-by-step probability solutions.

Probability of Two Independent Events

0 ≤ value ≤ 1
0 ≤ value ≤ 1

What Is Probability?

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). A probability of 0.25 means the event occurs in roughly a quarter of all trials over the long run; 0.95 means it occurs almost every time but isn't guaranteed. The same number can be expressed as a decimal (0.25), a percentage (25%), odds (1 in 4), or a fraction (1/4) — all four formats appear on every result card above.

This calculator covers the four most-asked probability questions in one place: combining two events with intersection, union, and exclusive-or; deducing missing probabilities from any two known values; chaining a single event across multiple independent trials; and finding the area under a normal distribution between any two bounds. Related tools include the percentage calculator, scientific calculator, and dice roller.

How Probability Calculation Works

Complement Rule

Every event A has a paired non-event A'. Their probabilities always sum to 1: P(A) + P(A') = 1. When the chance of an event is easier to compute, subtract from 1 for its complement.

Independence

Two events are independent when knowing whether one happened tells you nothing about the other. For independent events, the joint probability is the product: P(A∩B) = P(A) · P(B).

Inclusion–Exclusion

The union P(A∪B) counts every outcome where A or B (or both) occur. The naive sum P(A) + P(B) double-counts the overlap, so we subtract it: P(A∪B) = P(A) + P(B) − P(A∩B).

Normal Distribution CDF

Continuous variables use a density curve f(x). Probability between two bounds is the area under that curve, computed from the cumulative distribution function Φ(z) via the error function.

6 Ways to Use This Probability Calculator

1

Combine two events

Find intersection, union, exclusive-or, and complement probabilities for any pair of independent events.

2

Reverse-solve unknowns

Provide any two known probabilities and let the solver derive the remaining six values automatically.

3

Project repeated trials

See how the "at least once" probability grows and the "all success" chance shrinks across multiple attempts.

4

Score the normal curve

Compute area under a bell curve between any two bounds (including ±∞), with full confidence-interval table.

5

Learn visually

Drag the explorer sliders and watch the Venn diagram update live to build genuine probability intuition.

6

Solve coin / dice / cards

Quick playgrounds for binomial coin flips, n-dice sums, and standard-deck card draw probabilities.

Best Practices When Working With Probability

Most probability errors come from confusing very similar quantities — union with intersection, independent with mutually exclusive, conditional with joint — or from forgetting that every event has a complement. Train yourself to write the symbol before plugging in numbers; the algebra usually fixes itself once the right identity is named.

Always check that probabilities sum to 1 across an exhaustive partition. The four cells "only A," "only B," "both," and "neither" carve the universe into mutually exclusive regions whose probabilities total exactly one. If yours don't, an identity was misapplied somewhere.

When applying the independence assumption to a real-world scenario, verify that knowing one outcome really doesn't change the other. Two card draws without replacement are not independent; two coin tosses are. Rain and traffic are correlated, not independent — even if you can still build a useful model by pretending otherwise.

Why Probability Matters

Risk & insurance

Insurance premiums, credit scoring, and reliability engineering all multiply small per-event probabilities across large portfolios to forecast expected loss.

Science & medicine

Clinical trials use p-values and confidence intervals — direct outputs of the normal distribution — to decide whether a treatment effect is real or noise.

Machine learning

Bayesian updating, softmax outputs, and dropout regularisation all live inside probability theory. Even "deterministic" models output probabilities.

Everyday decisions

Weather forecasts, sports analytics, game strategy, and even traffic-light timing rest on probability estimates a lay person rarely has to compute manually — but always has to interpret.

Where Probability Gets Tricky

Independent vs mutually exclusive

Two events can be neither independent nor mutually exclusive, both, or just one. They are different concepts: mutual exclusion means P(A∩B)=0, independence means P(A∩B)=P(A)·P(B). The only way both hold is when at least one event has zero probability.

Conditional vs joint

P(A|B) reads "A given B" — the chance of A in the subset of outcomes where B already happened. P(A∩B) is the chance both happen together. They differ by a factor of P(B): P(A∩B) = P(A|B) · P(B).

Base-rate fallacy

A 99% accurate cancer test in a population where 1% have the disease yields more false positives than true positives. Always weight conditional probabilities by the prior — Bayes' theorem makes this explicit.

Gambler's fallacy

Past independent outcomes never "owe" you a future result. After ten reds at roulette, the next spin is still 18/38 black — the wheel has no memory.

Confusing odds with probability

Odds of 2:1 = probability 2/3, not 2/1. Probability is event ÷ all outcomes; odds is event ÷ non-event. The result cards on this page show both for the same value.

Normal distribution misuse

Not every dataset is normal. Income, web-traffic, and earthquake magnitudes are heavy-tailed — using a normal CDF on them under-estimates extreme events by orders of magnitude.

The Core Probability Formulas

Complement

P(A') = 1 − P(A)

Sums to 1.

Joint (indep.)

P(A ∩ B) = P(A) · P(B)

Only under independence.

Union

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Inclusion–exclusion.

Symmetric diff

P(A Δ B) = P(A) + P(B) − 2 P(A ∩ B)

Exactly one occurs.

Neither

P((A ∪ B)') = 1 − P(A ∪ B)

Complement of the union.

Only A

P(A ∩ B') = P(A) − P(A ∩ B)

A happens but not B.

Series of n

P(at least once) = 1 − (1 − p)^n

Repeated independent trials.

Binomial

P(k of n) = C(n,k) · p^k · (1−p)^(n−k)

Exactly k successes.

Normal PDF

f(x) = 1 / (σ√2π) · e^(−(x−μ)² / 2σ²)

Bell curve density.

Z-score

z = (x − μ) / σ

Standardise any value.

Normal CDF

P(X ≤ x) = Φ(z) = ½ (1 + erf(z / √2))

Area under the curve.

Conditional

P(A | B) = P(A ∩ B) / P(B)

A given B occurred.

Common Probability Mistakes

  1. 1

    Adding when you should multiply

    "Two 50/50 coins gives 100%" — no. Independent events multiply for joint probability: 0.5 × 0.5 = 0.25 (HH). The 100% claim conflates union with intersection.

  2. 2

    Using independence on dependent events

    Drawing two aces without replacement is 4/52 × 3/51 — not (4/52)². The second draw depends on the first.

  3. 3

    Forgetting the "or both" in union

    "A or B" in everyday speech usually means exclusive or. In probability, P(A∪B) includes the both-occur case. Use P(AΔB) when you want strict either-or.

  4. 4

    Treating odds and probability as interchangeable

    Odds of 3:1 = probability 0.75, not 0.30. Always confirm the units of any quoted "odds."

  5. 5

    Misreading a confidence interval

    A 95% CI does not mean "there's a 95% chance the true value lies in the interval." It means "95% of intervals built this way will contain the true value." The distinction matters in statistics journals.

  6. 6

    Plugging probabilities outside [0, 1]

    Any negative answer, or anything bigger than 1, is a sign of an algebra error. The solver here checks both bounds and flags inconsistencies.

The Normal Distribution Explained

The normal distribution (also called the Gaussian distribution or bell curve) is the most important continuous probability distribution in statistics. It arises naturally whenever a quantity is the sum of many small independent influences — the central limit theorem guarantees that the average of many independent samples is approximately normal regardless of the underlying distribution.

Two parameters fully describe a normal distribution: the mean μ (centre of the bell) and the standard deviation σ (spread). The famous 68–95–99.7 rule says that ~68% of values fall within μ ± 1σ, ~95% within μ ± 2σ, and ~99.7% within μ ± 3σ. The Normal tab on this page generates the exact percentages for any bounds you choose.

To compute the probability of falling in a specific range, we standardise to z-scores (z = (x − μ) / σ) and look up the cumulative density Φ(z). This calculator uses the Abramowitz & Stegun error-function approximation, which is accurate to better than 1.5×10⁻⁷ across the full range — more than enough for any teaching or applied use.

Built for students, statisticians, data scientists, and curious minds.

Probability formulas and the normal-distribution numerical approximations are sourced from standard references including Abramowitz & Stegun's Handbook of Mathematical Functions, Sheldon Ross's A First Course in Probability, and NIST's Engineering Statistics Handbook. See our methodology and editorial policy. Educational only — confidence intervals and p-values are not a substitute for domain expertise.

Frequently Asked Questions

Probability is the mathematical measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). A probability of 0.25 means the event happens roughly one quarter of the time over the long run, while 0.95 means it happens almost every time but is not guaranteed. The same value can be expressed as a decimal (0.25), a percentage (25%), odds (1 in 4), or a fraction (1/4) — every result card on this page shows all four formats so you can use the one that matches your context.

For a single event with equally likely outcomes, probability = number of favourable outcomes ÷ total number of outcomes. For example, the chance of rolling a 4 on a fair six-sided die is 1/6 ≈ 0.167. For two independent events, multiply for the joint probability (P(A∩B) = P(A) · P(B)) and use inclusion–exclusion for the union (P(A∪B) = P(A) + P(B) − P(A∩B)). The Two Events tab on this page automates both formulas and shows every related probability — complement, exclusive OR, neither, only A, only B — in a single result card grid.

Two events are independent when knowing whether one happened tells you nothing about whether the other will happen. Coin flips are independent — the second flip is still 50/50 regardless of the first. Drawing two cards without replacing the first is NOT independent, because the second draw is constrained by the first. The mathematical test is P(A∩B) = P(A) · P(B): if the joint probability equals the product of the individual probabilities, the events are independent. This calculator assumes independence between A and B unless you switch to a conditional-probability tool like a Bayes Theorem Calculator.

The union of two events A and B — written P(A ∪ B) — is the probability that A happens, B happens, or both happen. It is calculated with the inclusion–exclusion principle: P(A∪B) = P(A) + P(B) − P(A∩B). The subtraction removes the overlap that would otherwise be double-counted. If A and B cannot happen together (mutually exclusive), P(A∩B) = 0 and the union simplifies to P(A) + P(B). The Two Events tab on this page computes the union automatically and highlights the corresponding region on the interactive Venn diagram.

The intersection of two events A and B — written P(A ∩ B) — is the probability that BOTH events occur. For independent events, this is the product of their individual probabilities: P(A∩B) = P(A) · P(B). Two coins both landing heads has probability 0.5 × 0.5 = 0.25. For dependent events, you must use the conditional form P(A∩B) = P(A) · P(B|A). The intersection is always ≤ either individual probability — adding events can only narrow the set of outcomes, not widen it.

The complement of event A — written P(A') or P(Aᶜ) — is the probability that A does NOT occur. It is always P(A') = 1 − P(A), because either A happens or A doesn't, and those two cases must add to certainty. Complements are useful when the "not" case is easier to compute than the direct case. For example, finding the probability of at least one head in five coin flips is awkward directly (you'd sum five binomial terms), but its complement — zero heads in five flips — is just (1/2)⁵, so the answer is 1 − 1/32 = 31/32.

The normal distribution (also called the Gaussian distribution or bell curve) is the most important continuous probability distribution in statistics. It describes any quantity that arises as the sum of many small independent influences — by the central limit theorem, the average of many independent samples is approximately normal regardless of the underlying distribution. A normal distribution is fully specified by its mean μ (centre) and standard deviation σ (spread). The famous 68–95–99.7 rule says ~68% of values fall within ±1σ, ~95% within ±2σ, and ~99.7% within ±3σ of the mean. The Normal tab on this page calculates exact probabilities for any bounds and any μ and σ.

A confidence interval is a range of values, calculated from data, that is likely to contain the true unknown parameter you are estimating. A 95% confidence interval means that if you repeated your sampling procedure many times and built an interval the same way each time, about 95% of those intervals would contain the true value. Note the subtle distinction: it is NOT a probability statement about the parameter itself, but about the procedure. For a normal distribution, a 95% confidence interval extends from μ − 1.96σ to μ + 1.96σ. The Normal tab generates the full table from 68.27% to 99.999%.

A z-score (also called a standard score) tells you how many standard deviations a value lies above or below the mean. The formula is z = (x − μ) / σ. A z-score of 1.5 means the value sits 1.5σ above the mean; a z-score of −2 sits 2σ below. Z-scores standardise any normal distribution to the standard normal (μ = 0, σ = 1), so the same probability tables apply universally. They are also the foundation of hypothesis testing, confidence intervals, and outlier detection. The Normal tab on this page shows the z-scores for both bounds alongside the probability calculation.

The discrete probability formulas — complement, intersection, union, exclusive OR — are exact, computed in double-precision IEEE 754 floating point with roughly 15 significant digits of precision. The normal distribution calculations use the Abramowitz & Stegun error-function approximation (formula 7.1.26), which is accurate to better than 1.5 × 10⁻⁷ across the full range of input — more than enough for any teaching, applied statistics, or industrial use case. The inverse-normal (probit) function used for the confidence-interval table uses the Wichura AS241 algorithm, also accurate to roughly seven significant digits. For mission-critical statistical work, cross-check against a statistical package like R, SciPy, or SAS.

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