Calculate Probability Calculator

Complete probability toolkit

Choose the model that matches your probability question

Move from a single sample space to two-event rules, reverse solving, repeated independent trials, or an area under a normal curve.

P(A) · P(A∩B) · P(A|B)Discrete events, repeated trials, and normal areasEach tool states its assumptions before it calculates.

How will you enter the single-event probability?

1Define the eventWhat counts as success?
2Count the sample spaceInclude every equally likely outcome.
3Read the resultCompare fraction, decimal, percent, and odds.

Verified formula 1.0.0Whole-number counts · client-side calculation · no forecast

Result

Your probability

The same event is shown in equivalent forms below.

Ready

Chance the event occurs50%

An even chance1/2 of the equally likely outcomes

Visual split

Event and complement

2 total

Event E occursFavorable outcomes
1 · 50%
Event E does not occurComplement outcomes
1 · 50%

Reduced fraction: 1/2
Decimal: 0.5
Percent: 50%
Complement: 50%
Odds in favor: 1:1
Odds against: 1:1

Where this result sits50%

Show the calculation

Check that the event is inside a positive sample space.0 ≤ 1 ≤ 2
Divide favorable outcomes by all equally likely outcomes.P(E) = 1 / 2 = 1/2 ≈ 0.5 = 50%
Use the remaining outcomes to calculate the complement.P(Eᶜ) = 1 / 2 ≈ 0.5 = 50%

Interpretation boundary

The count formula is valid only when the listed elementary outcomes are equally likely. Counts alone do not prove that assumption.

Two-event model

Independent, overlapping, or mutually exclusive

Independent, overlapping, or mutually exclusive events use different intersection rules. Choose the relationship first, then inspect every related probability.

P(A∪B) = P(A)+P(B)−P(A∩B)

Quick picks · fill and calculate

Result

Complete event relationship

P(A∪B) · at least one70%0.7

A only · 30%Both · 20%B only · 20%Neither · 30%

Event A50%P(A) 0.51/2

Event B40%P(B) 0.42/5

Not A50%P(Aᶜ) 0.51/2

Not B60%P(Bᶜ) 0.63/5

A and B20%P(A∩B) 0.21/5

A or B or both70%P(A∪B) 0.77/10

Exactly one50%P(A△B) 0.51/2

Neither30%P((A∪B)ᶜ) 0.33/10

A but not B30%P(A∩Bᶜ) 0.33/10

B but not A20%P(Aᶜ∩B) 0.21/5

A given B50%P(A|B) 0.51/2

B given A40%P(B|A) 0.42/5

Step-by-step formulas

P(A∩B) = 0.5 × 0.4 = 0.2.
P(A∪B) = 0.5 + 0.4 − 0.2 = 0.7.
P(A△B) = P(A∪B) − P(A∩B) = 0.5.
P(neither) = 1 − 0.7 = 0.3.
P(A|B) = 0.2 ÷ 0.4 = 0.5.

Reverse probability solver

Solve an independent two-event model

Choose exactly two of eight related probabilities. The solver assumes A and B are independent, then reports whether the information gives one labeled model, two swapped labels, too little information, or a contradiction.

P(A∩B)=P(A)P(B)

Result

Solver classification

uniqueOne labeled solution satisfies both values.

One direct event value and one independent identity determine the other event.

Recovered model

Event A50%P(A) 0.5

Event B80%P(B) 0.8

Not A50%P(Aᶜ) 0.5

Not B20%P(Bᶜ) 0.2

A and B40%P(A∩B) 0.4

A or B or both90%P(A∪B) 0.9

Exactly one50%P(A△B) 0.5

Neither10%P((A∪B)ᶜ) 0.1

A but not B10%P(A∩Bᶜ) 0.1

B but not A40%P(Aᶜ∩B) 0.4

Repeated independent trials

Probability of a series of independent events

Enter a per-trial probability and repeat count for A and B. Trials must keep the same probability and remain independent.

P(at least once)=1−(1−p)ⁿ

Result

All, none, and at least once

Event A · 5 trials

All 5: 0.07776 · 7.776%
None: 0.01024 · 1.024%
At least once: 0.98976 · 98.976%

Event B · 3 trials

All 3: 0.027 · 2.7%
None: 0.343 · 34.3%
At least once: 0.657 · 65.7%

Every A and every B succeeds0.002099520.209952%pAⁿ × pBᵐ

No A and no B success0.003512320.351232%(1−pA)ⁿ × (1−pB)ᵐ

At least one A and at least one B0.6502723265.027232%[1−(1−pA)ⁿ][1−(1−pB)ᵐ]

Every A, but no B0.026671682.667168%pAⁿ(1−pB)ᵐ

Every B, but no A0.000276480.027648%pBᵐ(1−pA)ⁿ

At least one A, but no B0.3394876833.948768%[1−(1−pA)ⁿ](1−pB)ᵐ

At least one B, but no A0.006727680.672768%[1−(1−pB)ᵐ](1−pA)ⁿ

Continuous probability

Probability of a normal distribution

Standardize two raw bounds and measure the area between them. Use -inf or inf for an open tail, then compare central confidence ranges on the same scale.

P(L<X<R)=Φ(zᵣ)−Φ(zₗ)

Common standard-normal areas

Result

Area under the normal curve

Probability between the bounds68.268949%0.6826894921

Shaded interval · 68.268949%

Between L and R68.268949%P(L<X<R) 0.6826894921

Outside the interval31.731051%P(X≤L or X≥R) 0.3173105079

Left tail15.865525%P(X≤L) 0.1586552539

Right tail15.865525%P(X≥R) 0.1586552539

Standardization steps

zₗ = (-1 − 0) / 1 = -1
zᵣ = (1 − 0) / 1 = 1
Φ(zᵣ) − Φ(zₗ) = 0.6826894921

Reference table

Central confidence intervals for N(μ, σ²)

Each row centers the stated normal-model coverage on μ and scales the critical z value by σ.

Central coverage	Critical z	Range
68.268949%	1	-1 to 1
80%	1.28155157	-1.28155157 to 1.28155157
90%	1.64485363	-1.64485363 to 1.64485363
95%	1.95996398	-1.95996398 to 1.95996398
98%	2.32634787	-2.32634787 to 2.32634787
99%	2.5758293	-2.5758293 to 2.5758293
99.5%	2.80703377	-2.80703377 to 2.80703377
99.8%	3.09023231	-3.09023231 to 3.09023231
99.9%	3.29052673	-3.29052673 to 3.29052673
99.99%	3.89059189	-3.89059189 to 3.89059189
99.999%	4.41717341	-4.41717341 to 4.41717341

Probability guide

From a sample space to a probability you can explain

The arithmetic is often short; choosing the right model is the real work. This guide connects sample spaces, AND/OR/conditional rules, reverse solving, repeated trials, and normal areas without hiding their assumptions.

Event: E ⊆ S
Probability: P(E) = |E| / |S|
Complement: P(Eᶜ) = 1 − P(E)

01 · Method

How to calculate probability from equally likely outcomes

The classical counting rule works only for a finite sample space whose elementary outcomes are complete, mutually exclusive, and equally likely.

1
Write the full sample space S
List every possible elementary outcome at one consistent level. For a fair die, S = {1, 2, 3, 4, 5, 6}. The total is |S| = 6.
2
Define the event E
Decide exactly what counts as success. For “roll an odd number,” E = {1, 3, 5}, so |E| = 3 favorable outcomes.
3
Divide, then interpret
Calculate |E| ÷ |S| = 3 ÷ 6 = 1/2. The decimal is 0.5 and the percentage is 50%. These are equivalent formats, not separate answers.

123456

Event EOdd faces: 1, 3, 53 outcomes
Complement EᶜEven faces: 2, 4, 63 outcomes

02 · Interpretation

One probability, four useful ways to write it

Fraction, decimal, percent, and odds emphasize different information. The calculator keeps them together so you do not have to translate between cards.

Fraction1/2

Decimal0.5

Percent50%

Odds in favor1:1

0Impossible0.25Unlikely0.5Even chance0.75Likely1Certain

Probability always lies from 0 through 1. Percentages run from 0% through 100%. Values near 1 are more likely, but “likely” never means guaranteed.

03 · Examples

Worked examples with the sample space shown

Each example identifies the denominator before dividing. That one habit prevents most classical-probability setup errors.

Example 1

Roll an odd number

Sample space: {1, 2, 3, 4, 5, 6}
Event: {1, 3, 5}

3 / 6 = 1 / 2 = 0.5 = 50%

A fair die makes all six faces equally likely.

Example 2

Draw an ace

Sample space: 52 cards
Event: 4 aces

4 / 52 = 1 / 13 ≈ 0.076923 = 7.6923%

The draw is from a complete, well-shuffled standard deck.

Example 3

Get two heads

Sample space: {HH, HT, TH, TT}
Event: {HH}

1 / 4 = 0.25 = 25%

HT and TH stay separate because they are distinct ordered outcomes.

04 · Assumptions

Equal likelihood is a modeling condition, not a counting trick

Classical count works

A fair six-sided die

The mechanism and “fair” assumption give each face the same probability, so counting three odd faces out of six is justified.

Counting labels is not enough

Six weather categories

Six labels do not imply a 1/6 chance for each category. Use observed data or a defensible probability model instead of favorable ÷ six.

Observed successes divided by observed trials is an empirical frequency. It can estimate a process rate, but it is not automatically a sample space of equally likely outcomes. Use the statistics calculator to inspect observed numeric data, and keep sampling uncertainty and study design in view.

05 · Related forms

Complement and odds come from the same two counts

ComplementP(Eᶜ) = 1 − P(E)

If 3 of 6 outcomes are favorable, the other 3 outcomes form the complement: P(Eᶜ) = 3/6 = 50%. Event plus complement always totals 1.

Odds in favorfavorable : unfavorable

For 4 aces among 52 cards, unfavorable outcomes are 48, so odds in favor are 4:48 = 1:12. Probability remains 4/52 = 1/13.

Odds againstunfavorable : favorable

Reverse the same counts. The ace example gives odds against of 48:4 = 12:1. Odds use a colon; probability uses a part-to-total fraction.

06 · Two events

AND, OR, exactly one, neither, and conditional probability

Two marginal probabilities do not determine their overlap unless the relationship is known. The Two events tool therefore asks whether A and B are independent, mutually exclusive, or have a supplied intersection.

Intersection · ANDP(A∩B) = P(A)P(B)

Use multiplication only for independent events. For dependent events, enter the known intersection or use P(A∩B) = P(A|B)P(B).

Union · inclusive ORP(A∪B) = P(A)+P(B)−P(A∩B)

Subtract the overlap once because adding P(A) and P(B) counts it twice. For mutually exclusive events the overlap is zero.

Exactly one · XORP(A△B) = P(A)+P(B)−2P(A∩B)

Exactly one combines A-only and B-only. It excludes both the overlap and the region where neither event occurs.

Neither eventP((A∪B)ᶜ) = 1−P(A∪B)

Neither is the complement of the entire union, not the same quantity as “not A” or “not B” considered separately.

A given B · conditionalP(A|B) = P(A∩B) / P(B)

Conditioning restricts the reference space to B, so P(B) must be greater than zero. Under independence, P(A|B) equals P(A).

Valid overlapmax(0, a+b−1) ≤ P(A∩B) ≤ min(a,b)

These Fréchet bounds prevent impossible Venn regions such as a negative neither probability or an overlap larger than either event.

Independent

Events can overlap without influencing each other

Two fair coin tosses are independent: heads on the first does not change the second, and P(H∩H) = 1/2 × 1/2 = 1/4.

Mutually exclusive

Events cannot happen together

One die roll cannot be both 1 and 2. Their overlap is zero and their probabilities must sum to no more than one. Except at zero-probability boundaries, mutually exclusive events are not independent.

07 · Reverse solver

How the independent two-event solver uses any informative pair

Under independence, eight displayed measures reduce to two unknowns, P(A) and P(B). The solver checks whether the chosen pair provides two independent equations before recovering the complete model.

Direct recoveryP(B) = P(A∩B) / P(A)

If P(A)=0.5 and P(A∩B)=0.4, then P(B)=0.8. The remaining complement, union, XOR, neither, A-only, and B-only values follow automatically.

Symmetric pairs = P(A)+P(B), t = P(A)P(B)

Intersection and union can determine s and t, then P(A) and P(B) are roots of x²−sx+t=0. The labels may be interchangeable, so both assignments are shown.

Information checkP(A∪B) + P(neither) = 1

Union and neither are the same fact in complementary form. Entering only those two values is redundant, while inconsistent complements are rejected.

08 · Repeated trials

All, none, and at least one success across independent trials

Repetition uses the same probability p on every trial and assumes trials are independent. Raising p to a power describes a specific all-success sequence; it is not the probability of exactly k successes unless k=n.

Success on every trialP(all n) = pⁿ

For p=0.6 repeated five times, P(all five)=0.6⁵=0.07776.

No successP(none) = (1−p)ⁿ

All trials must fall in the complement. With p=0.6 and n=5, this is 0.4⁵=0.01024.

At least one successP(at least one) = 1−(1−p)ⁿ

The complement shortcut avoids adding the probabilities of one, two, three, and more successes separately.

The Repeated trials tool also combines two independent series. For A with p=0.6 over five trials and B with p=0.3 over three trials, the probability that every A and every B succeeds is 0.6⁵×0.3³=0.00209952. If probabilities change between trials or sampling occurs without replacement, this fixed-p model is not valid.

09 · Normal distribution

Standardize the bounds, then measure the shaded normal area

For X∼N(μ,σ²), each raw bound is converted to a z-score. The interval probability is the difference between the standard-normal cumulative areas.

Standardizez = (x−μ) / σ

Mean and bounds must share a unit, and σ must be positive. A bound of −∞ maps to cumulative area 0; +∞ maps to area 1.

Between two boundsP(L<X<R) = Φ(zᵣ)−Φ(zₗ)

For the standard normal curve between −1 and 1, the area is about 0.682689, or 68.2689%.

Outside and tailsP(outside) = Φ(zₗ)+1−Φ(zᵣ)

The left tail, interval, and right tail partition the full area and must total one, apart from final display rounding.

Central coverage	Critical z	Interval for N(μ,σ²)
68.27%	1.00000	`μ ± 1.00000σ`
90%	1.64485	`μ ± 1.64485σ`
95%	1.95996	`μ ± 1.95996σ`
99%	2.57583	`μ ± 2.57583σ`

A confidence interval and a probability interval are not automatically the same inferential statement. The calculator reports central ranges of the declared normal model; it does not estimate μ or σ from data and does not establish normality.

10 · Model choice

Choose the rule that matches the probability question

The page provides several models, but no one formula fits every question. Choose the matching tool and check its assumptions before trusting the arithmetic.

Question	Information needed	Typical rule
One event in an equally likely sample space	Favorable and total outcome counts	`P(E) = \|E\| / \|S\|`
Event does not occur	The event probability	`P(Eᶜ) = 1 − P(E)`
A or B occurs	P(A), P(B), and their overlap	`P(A∪B) = P(A)+P(B)−P(A∩B)`
A given B	Intersection and a nonzero P(B)	`P(A\|B) = P(A∩B) / P(B)`
At least one success in repeated trials	Trial probability, count, and independence	`1 − (1−p)ⁿ`
Area under a normal curve	Mean, standard deviation, and bounds	`Φ(zᵣ) − Φ(zₗ)`

Independence and mutual exclusivity are not synonyms. Independent events can occur together and satisfy P(A∩B) = P(A)P(B). Mutually exclusive events cannot occur together and have P(A∩B) = 0. For standardized positions, use the z-score calculator only when a defensible mean and positive standard deviation are available.

11 · Quality check

Common mistakes to catch before calculating

Using an incomplete denominator

The total must include every elementary outcome, including outcomes that do not satisfy the event.

Counting overlapping outcomes twice

Elementary outcomes in the sample space must be mutually exclusive. If two descriptions overlap, split them into non-overlapping cases first.

Assuming equal likelihood from labels

Six named categories do not automatically have probability 1/6 each. The process must justify equal weights.

Confusing odds with probability

A probability of 1/4 corresponds to odds in favor of 1:3, not odds of 1:4.

12 · FAQ

Questions about probability models and results

Can I enter a decimal probability such as 0.35?

Yes. In the Single event or Two events tool, choose direct probability entry and type 0.35, 35%, or an equivalent fraction such as 7/20. Choose favorable/total only when you actually have a finite outcome-count model.

Why must favorable outcomes be no greater than the total?

The event is a subset of the sample space. It can contain none, some, or all of the elementary outcomes, but it cannot contain more outcomes than the sample space itself.

What happens at probability 0 or 1?

Zero favorable outcomes gives an impossible event within the declared sample space. Favorable equal to total gives a certain event. These are valid boundaries, but the total must still be positive.

Is independence the same as equal likelihood?

No. Equal likelihood compares the weights of elementary outcomes in one sample space. Independence describes two events whose occurrence does not change each other’s probability. They answer different questions.

Can this result predict what will happen next?

No. It describes the declared theoretical sample space. A 25% probability does not guarantee one success in every four trials, and observed frequencies can vary in finite samples.

Why can the two-value solver return two labeled answers?

Some pairs, such as intersection and union, determine the two event probabilities but not which root should be named A and which should be named B. The solver shows both label assignments instead of inventing information that was not supplied.

What does repeated independently mean?

It means every trial keeps the same per-trial probability and one trial does not change another. Under that model, all n successes have probability pⁿ and no success has probability (1−p)ⁿ.

When is the normal-distribution result appropriate?

Use it only when a continuous variable is reasonably modeled by a normal distribution with a defensible mean and positive standard deviation. The shaded area is a model probability, not proof that observed data are normal.

References

Formula and terminology sources

OpenStax Precalculus — Probability
OpenStax Introductory Statistics — Independent and Mutually Exclusive Events
NIST/SEMATECH — Normal Distribution
Use the fraction calculator to reduce an already validated event fraction.

Important limitations

Calculate Probability Calculator is limited to the finite inputs, modes, and outputs stated for the Calculate probability calculator method. Check favorable outcomes and total equally likely outcomes, validation messages, the equation, and the stated display precision before relying on event probability and complement probability.

Review the stated scope, calculation method, assumptions, supported inputs, and effective dates, and obtain qualified advice before using a high-stakes result.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Probability guide

From a sample space to a probability you can explain

Event: E ⊆ S
Probability: P(E) = |E| / |S|
Complement: P(Eᶜ) = 1 − P(E)

01 · Method

How to calculate probability from equally likely outcomes

The classical counting rule works only for a finite sample space whose elementary outcomes are complete, mutually exclusive, and equally likely.

1
Write the full sample space S
List every possible elementary outcome at one consistent level. For a fair die, S = {1, 2, 3, 4, 5, 6}. The total is |S| = 6.
2
Define the event E
Decide exactly what counts as success. For “roll an odd number,” E = {1, 3, 5}, so |E| = 3 favorable outcomes.
3
Divide, then interpret
Calculate |E| ÷ |S| = 3 ÷ 6 = 1/2. The decimal is 0.5 and the percentage is 50%. These are equivalent formats, not separate answers.

123456

Event EOdd faces: 1, 3, 53 outcomes
Complement EᶜEven faces: 2, 4, 63 outcomes

02 · Interpretation

One probability, four useful ways to write it

Fraction, decimal, percent, and odds emphasize different information. The calculator keeps them together so you do not have to translate between cards.

Fraction1/2

Decimal0.5

Percent50%

Odds in favor1:1

0Impossible0.25Unlikely0.5Even chance0.75Likely1Certain

Probability always lies from 0 through 1. Percentages run from 0% through 100%. Values near 1 are more likely, but “likely” never means guaranteed.

03 · Examples

Worked examples with the sample space shown

Each example identifies the denominator before dividing. That one habit prevents most classical-probability setup errors.

Example 1

Roll an odd number

Sample space: {1, 2, 3, 4, 5, 6}
Event: {1, 3, 5}

3 / 6 = 1 / 2 = 0.5 = 50%

A fair die makes all six faces equally likely.

Example 2

Draw an ace

Sample space: 52 cards
Event: 4 aces

4 / 52 = 1 / 13 ≈ 0.076923 = 7.6923%

The draw is from a complete, well-shuffled standard deck.

Example 3

Get two heads

Sample space: {HH, HT, TH, TT}
Event: {HH}

1 / 4 = 0.25 = 25%

HT and TH stay separate because they are distinct ordered outcomes.

04 · Assumptions

Equal likelihood is a modeling condition, not a counting trick

Classical count works

A fair six-sided die

The mechanism and “fair” assumption give each face the same probability, so counting three odd faces out of six is justified.

Counting labels is not enough

Six weather categories

Six labels do not imply a 1/6 chance for each category. Use observed data or a defensible probability model instead of favorable ÷ six.

05 · Related forms

Complement and odds come from the same two counts

ComplementP(Eᶜ) = 1 − P(E)

If 3 of 6 outcomes are favorable, the other 3 outcomes form the complement: P(Eᶜ) = 3/6 = 50%. Event plus complement always totals 1.

Odds in favorfavorable : unfavorable

For 4 aces among 52 cards, unfavorable outcomes are 48, so odds in favor are 4:48 = 1:12. Probability remains 4/52 = 1/13.

Odds againstunfavorable : favorable

Reverse the same counts. The ace example gives odds against of 48:4 = 12:1. Odds use a colon; probability uses a part-to-total fraction.

06 · Two events

AND, OR, exactly one, neither, and conditional probability

Intersection · ANDP(A∩B) = P(A)P(B)

Use multiplication only for independent events. For dependent events, enter the known intersection or use P(A∩B) = P(A|B)P(B).

Union · inclusive ORP(A∪B) = P(A)+P(B)−P(A∩B)

Subtract the overlap once because adding P(A) and P(B) counts it twice. For mutually exclusive events the overlap is zero.

Exactly one · XORP(A△B) = P(A)+P(B)−2P(A∩B)

Exactly one combines A-only and B-only. It excludes both the overlap and the region where neither event occurs.

Neither eventP((A∪B)ᶜ) = 1−P(A∪B)

Neither is the complement of the entire union, not the same quantity as “not A” or “not B” considered separately.

A given B · conditionalP(A|B) = P(A∩B) / P(B)

Conditioning restricts the reference space to B, so P(B) must be greater than zero. Under independence, P(A|B) equals P(A).

Valid overlapmax(0, a+b−1) ≤ P(A∩B) ≤ min(a,b)

These Fréchet bounds prevent impossible Venn regions such as a negative neither probability or an overlap larger than either event.

Independent

Events can overlap without influencing each other

Two fair coin tosses are independent: heads on the first does not change the second, and P(H∩H) = 1/2 × 1/2 = 1/4.

Mutually exclusive

Events cannot happen together

One die roll cannot be both 1 and 2. Their overlap is zero and their probabilities must sum to no more than one. Except at zero-probability boundaries, mutually exclusive events are not independent.

07 · Reverse solver

How the independent two-event solver uses any informative pair

Under independence, eight displayed measures reduce to two unknowns, P(A) and P(B). The solver checks whether the chosen pair provides two independent equations before recovering the complete model.

Direct recoveryP(B) = P(A∩B) / P(A)

If P(A)=0.5 and P(A∩B)=0.4, then P(B)=0.8. The remaining complement, union, XOR, neither, A-only, and B-only values follow automatically.

Symmetric pairs = P(A)+P(B), t = P(A)P(B)

Intersection and union can determine s and t, then P(A) and P(B) are roots of x²−sx+t=0. The labels may be interchangeable, so both assignments are shown.

Information checkP(A∪B) + P(neither) = 1

Union and neither are the same fact in complementary form. Entering only those two values is redundant, while inconsistent complements are rejected.

08 · Repeated trials

All, none, and at least one success across independent trials

Success on every trialP(all n) = pⁿ

For p=0.6 repeated five times, P(all five)=0.6⁵=0.07776.

No successP(none) = (1−p)ⁿ

All trials must fall in the complement. With p=0.6 and n=5, this is 0.4⁵=0.01024.

At least one successP(at least one) = 1−(1−p)ⁿ

The complement shortcut avoids adding the probabilities of one, two, three, and more successes separately.

09 · Normal distribution

Standardize the bounds, then measure the shaded normal area

For X∼N(μ,σ²), each raw bound is converted to a z-score. The interval probability is the difference between the standard-normal cumulative areas.

Standardizez = (x−μ) / σ

Mean and bounds must share a unit, and σ must be positive. A bound of −∞ maps to cumulative area 0; +∞ maps to area 1.

Between two boundsP(L<X<R) = Φ(zᵣ)−Φ(zₗ)

For the standard normal curve between −1 and 1, the area is about 0.682689, or 68.2689%.

Outside and tailsP(outside) = Φ(zₗ)+1−Φ(zᵣ)

The left tail, interval, and right tail partition the full area and must total one, apart from final display rounding.

Central coverage	Critical z	Interval for N(μ,σ²)
68.27%	1.00000	`μ ± 1.00000σ`
90%	1.64485	`μ ± 1.64485σ`
95%	1.95996	`μ ± 1.95996σ`
99%	2.57583	`μ ± 2.57583σ`

10 · Model choice

Choose the rule that matches the probability question

The page provides several models, but no one formula fits every question. Choose the matching tool and check its assumptions before trusting the arithmetic.

Question	Information needed	Typical rule
One event in an equally likely sample space	Favorable and total outcome counts	`P(E) = \|E\| / \|S\|`
Event does not occur	The event probability	`P(Eᶜ) = 1 − P(E)`
A or B occurs	P(A), P(B), and their overlap	`P(A∪B) = P(A)+P(B)−P(A∩B)`
A given B	Intersection and a nonzero P(B)	`P(A\|B) = P(A∩B) / P(B)`
At least one success in repeated trials	Trial probability, count, and independence	`1 − (1−p)ⁿ`
Area under a normal curve	Mean, standard deviation, and bounds	`Φ(zᵣ) − Φ(zₗ)`

11 · Quality check

Common mistakes to catch before calculating

Using an incomplete denominator

The total must include every elementary outcome, including outcomes that do not satisfy the event.

Counting overlapping outcomes twice

Elementary outcomes in the sample space must be mutually exclusive. If two descriptions overlap, split them into non-overlapping cases first.

Assuming equal likelihood from labels

Six named categories do not automatically have probability 1/6 each. The process must justify equal weights.

Confusing odds with probability

A probability of 1/4 corresponds to odds in favor of 1:3, not odds of 1:4.

12 · FAQ

Questions about probability models and results

Can I enter a decimal probability such as 0.35?

Why must favorable outcomes be no greater than the total?

The event is a subset of the sample space. It can contain none, some, or all of the elementary outcomes, but it cannot contain more outcomes than the sample space itself.

What happens at probability 0 or 1?

Zero favorable outcomes gives an impossible event within the declared sample space. Favorable equal to total gives a certain event. These are valid boundaries, but the total must still be positive.

Is independence the same as equal likelihood?

Can this result predict what will happen next?

No. It describes the declared theoretical sample space. A 25% probability does not guarantee one success in every four trials, and observed frequencies can vary in finite samples.

Why can the two-value solver return two labeled answers?

What does repeated independently mean?

It means every trial keeps the same per-trial probability and one trial does not change another. Under that model, all n successes have probability pⁿ and no success has probability (1−p)ⁿ.

When is the normal-distribution result appropriate?

References

Formula and terminology sources

OpenStax Precalculus — Probability
OpenStax Introductory Statistics — Independent and Mutually Exclusive Events
NIST/SEMATECH — Normal Distribution
Use the fraction calculator to reduce an already validated event fraction.

Describe one sample space

Probability of event A

Equivalent forms

Define A, B, and their relationship

Complete event relationship

Provide any two informative measures

Solver classification

Recovered model

Two repeatable events

All, none, and at least once

Distribution and interval

Area under the normal curve

Central confidence intervals for N(μ, σ²)

How to calculate probability from equally likely outcomes

Write the full sample space S

Define the event E

Divide, then interpret

One probability, four useful ways to write it

Worked examples with the sample space shown

Roll an odd number

Draw an ace

Get two heads

Equal likelihood is a modeling condition, not a counting trick

A fair six-sided die

Six weather categories

Complement and odds come from the same two counts

AND, OR, exactly one, neither, and conditional probability

Events can overlap without influencing each other

Events cannot happen together

How the independent two-event solver uses any informative pair

All, none, and at least one success across independent trials

Standardize the bounds, then measure the shaded normal area

Choose the rule that matches the probability question

Common mistakes to catch before calculating

Using an incomplete denominator

Counting overlapping outcomes twice

Assuming equal likelihood from labels

Confusing odds with probability

Questions about probability models and results

Formula and terminology sources

Important limitations

Describe one sample space

Probability of event A

Equivalent forms

Define A, B, and their relationship

Complete event relationship

Provide any two informative measures

Solver classification

Recovered model

Two repeatable events

All, none, and at least once

Distribution and interval

Area under the normal curve

Central confidence intervals for N(μ, σ²)

How to calculate probability from equally likely outcomes

Write the full sample space S

Define the event E

Divide, then interpret

One probability, four useful ways to write it

Worked examples with the sample space shown

Roll an odd number

Draw an ace

Get two heads

Equal likelihood is a modeling condition, not a counting trick

A fair six-sided die

Six weather categories

Complement and odds come from the same two counts

AND, OR, exactly one, neither, and conditional probability

Events can overlap without influencing each other

Events cannot happen together

How the independent two-event solver uses any informative pair

All, none, and at least one success across independent trials

Standardize the bounds, then measure the shaded normal area

Choose the rule that matches the probability question

Common mistakes to catch before calculating

Using an incomplete denominator

Counting overlapping outcomes twice

Assuming equal likelihood from labels

Confusing odds with probability

Questions about probability models and results