Probability Calculator
Three modes: basic probability (P(A), P(A∩B), P(A∪B), conditional), permutations and combinations, and binomial distribution. Instant results with formulas.
Option 1 — Favorable / Total Outcomes
Option 2 — Two Events P(A) and P(B)
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How It Works
Enter favorable outcomes and total outcomes. Result shows P as fraction, decimal, and percentage. Also shows complementary probability P(not A) = 1−P(A). For combined events: enter P(A) and P(B) to compute P(A∩B) for independent events and P(A∪B) using the addition rule.
P(A) = favorable/total | P(not A) = 1 − P(A)P(head) = 1/2 = 0.5 = 50% | P(6 on die) = 1/6 ≈ 16.7%Enter n (total items) and r (items selected). Permutation P(n,r) = n!/(n−r)!: ordered selections (race placements, passwords, seating arrangements). Combination C(n,r) = n!/(r!(n−r)!): unordered (committees, card hands, lottery picks). Both compute using BigInt for large values.
P(n,r) = n!/(n−r)! | C(n,r) = n!/(r!(n−r)!)P(10,3)=720 | C(10,3)=120 | C(52,5)=2,598,960Enter: n (trials), k (successes wanted), p (probability of success). P(exactly k) = C(n,k) × p^k × (1−p)^(n−k). Also shows cumulative P(at most k) and P(at least k). Expected value = n×p. Variance = n×p×(1−p). Example: P(exactly 3 heads in 10 flips) = C(10,3)×0.5³×0.5⁷ = 120×(1/128)×(1/128) ≈ 11.7%.
P(X=k) = C(n,k)×p^k×(1−p)^(n−k)n=10, k=3, p=0.5 → P=11.72% | E[X]=5, Var=2.5P(A∪B) = P(A) + P(B) − P(A∩B). The subtraction removes double-counting of outcomes in both events. For mutually exclusive events (cannot both occur): P(A∩B)=0, so P(A∪B)=P(A)+P(B). Enter P(A), P(B), and P(A∩B) in Basic mode to compute.
P(A∪B) = P(A) + P(B) − P(A∩B)P(A)=0.3, P(B)=0.4, P(A∩B)=0.1 → P(A∪B)=0.6P(not A) = 1 − P(A). Often easier to compute P(no success) and subtract from 1. Example: P(at least one head in 3 flips) = 1 − P(no heads) = 1 − (0.5)³ = 1 − 0.125 = 0.875. In Binomial mode: P(at least k) = 1 − P(at most k−1) shows automatically in results.
P(not A) = 1 − P(A) | P(≥1) = 1 − P(0)P(≥1 head in 3 flips) = 1 − (0.5)³ = 0.875Quick Reference
Verify these in the calculator above.
Basic
P(head)
1/2 = 50%
Basic
P(6 on die)
1/6 ≈ 16.7%
Addition
P(A∪B): 0.3+0.4-0.1
0.6 = 60%
Complement
P(not 6 on die)
5/6 ≈ 83.3%
Permutation
P(10,3) permutations
720
Combination
C(10,3) combinations
120
Combination
C(52,5) poker hands
2,598,960
Binomial
Binomial n=10,k=3,p=0.5
≈11.72%
Tips & Shortcuts
For "at least one success" in n trials: use complementary rule — P(≥1) = 1 − P(none) = 1 − (1−p)^n. This is faster than summing individual probabilities.
Permutation vs combination: ask "does order matter?" Yes → permutation. No → combination. Committee of 3 from 10: order doesn't matter → C(10,3)=120.
In Binomial mode, the cumulative probability P(at most k) shows alongside P(exactly k) — useful for quality control (what fraction of items have at most k defects?).
P(A) × P(B) = P(A∩B) only for INDEPENDENT events. For dependent events (like drawing cards without replacement), use conditional probability.
Check your answer: all probabilities must be between 0 and 1. If you get a value outside this range, check the inputs.
Common Mistakes
Multiplying probabilities for dependent events
P(drawing two aces without replacement): P(1st ace)=4/52, P(2nd ace|1st ace)=3/51. P(both)=(4/52)×(3/51)≠(4/52)². Dependency reduces the second probability.
Using P(A) + P(B) for overlapping events
If events can both occur, P(A∪B) = P(A)+P(B)−P(A∩B). Forgetting the subtraction double-counts overlapping outcomes. Only for mutually exclusive events is P(A∪B)=P(A)+P(B).
Confusing permutation with combination
P(5,3)=60 (ordered: ABC≠BAC). C(5,3)=10 (unordered: ABC=BAC). If order matters, permutation; otherwise combination. Committees, hands, groups → combination. Rankings, passwords, arrangements → permutation.
Entering probabilities greater than 1
Probabilities must be between 0 and 1. If the favorable outcomes exceed total outcomes, or P(A)+P(B) input exceeds 1, the calculation is invalid. Check that your inputs represent valid probabilities.
Applying binomial formula when trials are not independent
Binomial distribution requires: fixed n, binary outcome (success/fail), constant p, independent trials. Drawing cards without replacement violates independence — use hypergeometric distribution instead.
Frequently Asked Questions
Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1 (or 0% to 100%). P=0 means impossible, P=1 means certain. Formula: P(event) = favorable outcomes / total outcomes.
P(A∩B) = P(A) × P(B) if A and B are independent events. For dependent events: P(A∩B) = P(A) × P(B|A). Example: rolling two dice and getting both 6: P=(1/6)×(1/6)=1/36.
P(A∪B) = P(A) + P(B) − P(A∩B). The subtraction avoids double-counting events in both A and B. For mutually exclusive events (P(A∩B)=0): P(A∪B) = P(A) + P(B).
P(A|B) = P(A∩B)/P(B) — probability of A given that B has occurred. Example: probability of drawing a heart from a deck given the card is red: P(heart|red) = P(heart∩red)/P(red) = (1/4)/(1/2) = 1/2.
Binomial distribution models n independent trials each with probability p of success. P(exactly k successes) = C(n,k) × p^k × (1−p)^(n−k). Example: probability of exactly 3 heads in 10 fair coin flips.
Permutation P(n,r) = n!/(n−r)!: ordered arrangements of r items from n. Combination C(n,r) = n!/(r!(n−r)!): unordered selections. For choosing a committee: order doesn't matter → combination. For race placement: order matters → permutation.
P(not A) = 1 − P(A). If P(rain tomorrow)=0.3, then P(no rain)=0.7. Useful when calculating the probability of "at least one" success: P(at least 1) = 1 − P(none).
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