What is P(A and B) for independent events?
Multiply: P(A and B) = P(A) × P(B). Example: two 50% events both occur with probability 0.25.
P(not A) = 0.500000
P(A and B) = 0.125000
P(A or B) = 0.625000
Combine independent event probabilities—multiply for AND, inclusion-exclusion for OR, complement for NOT—with decimal and percent I/O.
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For independent A and B: P(A and B) = P(A)×P(B). P(A or B) = P(A) + P(B) − P(A)×P(B). Complement: P(not A) = 1 − P(A). Example: two fair coins both heads = 0.25.
P(not A) = 0.500000
P(A and B) = 0.125000
P(A or B) = 0.625000
Probability assigns a number from 0 (never) to 1 (always) to an uncertain outcome. This calculator focuses on **independent** events—knowing A happened does not change the chance of B.
**AND (both happen):** multiply. Flip a fair coin (P = 0.5) and roll a die for a 6 (P = 1/6 ≈ 0.167). Both together: 0.5 × 0.167 ≈ 0.083.
**OR (at least one):** add, then subtract the overlap counted twice: P(A or B) = P(A) + P(B) − P(A)×P(B). For mutually exclusive events the overlap is zero, so OR simplifies to addition.
**NOT A:** complements sum to 1. If rain chance is 0.35, dry chance is 0.65.
Counting problems start on combination and permutation calculators—favorable outcomes ÷ total outcomes. This page assumes you already converted counts to decimals or percents.
Multiply for independent AND; inclusion-exclusion for OR; subtract from 1 for complements.
Concise answers for common searches — definitions, steps, and comparisons.
Multiply: P(A and B) = P(A) × P(B). Example: two 50% events both occur with probability 0.25.
P(A or B) = P(A) + P(B) − P(A)×P(B) for independent events. Mutually exclusive: just add.
P(not A) = 1 − P(A). If success chance is 0.8, failure chance is 0.2.
1 − (0.9)² = 0.19, or 0.1 + 0.1 − 0.01 = 0.19 via inclusion-exclusion.
Accepts P(A) and optional P(B) as decimals or percents, validates range [0,1], and applies standard independent-event formulas.
Formula
P(not A) = 1 − P(A). Independent: P(A∩B) = P(A)P(B). P(A∪B) = P(A) + P(B) − P(A)P(B). For mutually exclusive events: P(A∪B) = P(A) + P(B).Convert “35%” to 0.35. The tool accepts either format.
Multiply only when outcomes do not influence each other—otherwise conditional methods apply.
Joint, union, and complement update live for two-event mode.
For equally likely outcomes, compute favorable/total on the combination calculator first, then enter the ratio here.
Input
P(heads) = 0.5 each flipOutput
P(both heads) = 0.25Independent: 0.5 × 0.5 = 0.25.
Input
P(rain) = 0.35Output
P(dry) = 0.651 − 0.35 = 0.65.
Input
P(A) = 0.4, P(B) = 0.3Output
P(A or B) = 0.580.4 + 0.3 − 0.12 = 0.58.
Input
P(roll 1) = 1/6, P(roll 2) = 1/6, exclusiveOutput
P(1 or 2) = 1/3No overlap—add: 2/6 = 1/3.
Input
P(A) = 0.02, P(B) = 0.03Output
P(A and B) = 0.0006Two rare independent events multiply to very small joint chance.
Input
1 royal flush in 52C5 handsOutput
P ≈ 1/2,598,960 ≈ 0.000000385Count favorable hands, divide by combination total, then analyze further events here.
Common real-world scenarios where this tool saves time.
Server A fails 2% of days, Server B 3%. Both fail same day: 0.02 × 0.03 = 0.0006 = 0.06%.
Independent 10% drop rates on two bosses: at least one drop = 0.1 + 0.1 − 0.01 = 0.19.
Two independent tests each catch 80% of defects: both catch same defect 0.64 of the time.
If P(success) = 0.73, verify P(failure) = 0.27 before submitting.
Step-by-step chains that connect related tools for common tasks.
| Rule | Formula | P(A)=0.4, P(B)=0.5 |
|---|---|---|
| Complement A | 1 − P(A) | 0.6 |
| A AND B | P(A)×P(B) | 0.20 |
| A OR B | P(A)+P(B)−P(A)P(B) | 0.70 |
| Neither | (1−P(A))(1−P(B)) | 0.30 |
| Experiment | Favorable | Total | P |
|---|---|---|---|
| Fair die, roll 6 | 1 | 6 | 1/6 ≈ 0.167 |
| Fair coin, heads | 1 | 2 | 0.5 |
| Card, ace | 4 | 52 | 4/52 ≈ 0.077 |
| Two dice sum 7 | 6 | 36 | 6/36 ≈ 0.167 |
Overlap region visualizes why union subtracts the intersection.
Quick sanity test after every complement calculation.
Poker and lottery odds start as combinations before becoming decimals.
Multiply in decimal form to avoid percent-vs-fraction confusion.
Independent AND means multiply. Adding is for mutually exclusive OR.
When events can both occur, subtract P(A)×P(B) after adding marginals.
Valid probability must lie between 0 and 1 inclusive.
Drawing two aces without replacement is not 4/52 × 4/52—use conditional probability.
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P(A|B) = P(A)—knowing B does not change A's chance. Coin flips are the textbook example.
When events are mutually exclusive—both cannot happen. Then P(A or B) = P(A) + P(B).
If each outcome is equally likely, P = favorable combinations / total combinations.
No. Values above 1 or below 0 indicate an error in setup or input.
1 − P(none) = 1 − ∏(1 − p_i). Often easier than inclusion-exclusion for many events.
Only when outcomes are defined as ordered. Use permutations for ordered favorable counts, combinations for unordered.
Ordered outcomes (e.g., die1=3, die2=5) count separately in sample spaces built from permutations of results.
Probability values stay local in your browser.
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Reviewed by EverydayTools Editorial Team on 2026-07-03.
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