Statistics · Probability
Probability
Calculator
Calculate probability of two events, binomial distributions, combinations, permutations, and odds conversions — with step-by-step solutions and visualizations.
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Probability of Two Events
💡 Enter P(A), P(B), and optionally P(A∩B). If A and B are independent, leave P(A∩B) blank.
0–1
0–1
0–1
Binomial Distribution
P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
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k
Combinations & Permutations
n
r
nCr = n! / (r!(n-r)!) — order does NOT matter
nPr = n! / (n-r)! — order DOES matter
nPr = n! / (n-r)! — order DOES matter
Odds & Probability Converter
Convert between probability, percentage, decimal odds, fractional odds, and "X in Y" form.
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Results
📊 Binomial Distribution
📐 Probability Rules Reference
Complement
P(A') = 1 − P(A)
If P(A)=0.3, P(A')=0.7
Addition Rule
P(A∪B) = P(A)+P(B)−P(A∩B)
Subtract overlap to avoid double counting
Mutual Exclusion
P(A∩B) = 0 → P(A∪B) = P(A)+P(B)
Cannot occur simultaneously
Independence
P(A∩B) = P(A) × P(B)
One event doesn't affect the other
Conditional Prob
P(A|B) = P(A∩B) / P(B)
Probability of A given B occurred
Bayes' Theorem
P(A|B) = P(B|A)×P(A) / P(B)
Update beliefs with new evidence
Combinations
nCr = n! / (r!(n-r)!)
Order doesn't matter
Permutations
nPr = n! / (n-r)!
Order matters
Understanding Probability
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). It's fundamental to statistics, decision-making, genetics, finance, and everyday reasoning.
Key Probability Concepts
P(A) Probability of event A (0 to 1)
P(A') Complement: probability A does NOT happen = 1−P(A)
P(A∩B) Intersection: BOTH A and B happen
P(A∪B) Union: AT LEAST ONE of A or B happens
P(A|B) Conditional: probability of A GIVEN B occurred
Independent: P(A∩B) = P(A)×P(B) [knowing B tells nothing about A]
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot both happen at the same time — P(A∩B)=0. For example, a coin flip: heads and tails are mutually exclusive. Independent events don't affect each other's probability — P(A|B)=P(A). For example, two separate coin flips: first flip doesn't affect second. Note: mutually exclusive events (with P>0) are actually DEPENDENT — if A happens, you know B didn't.
How does the binomial distribution work?
The binomial distribution models the number of successes in n independent trials, each with probability p. Formula: P(X=k) = C(n,k) × p^k × (1-p)^(n-k). The mean (expected successes) is n×p, and standard deviation is √(n×p×(1-p)). For example, flipping a coin 10 times: P(exactly 5 heads) = C(10,5) × 0.5^5 × 0.5^5 = 252 × (1/1024) ≈ 24.6%.
When do I use combinations vs permutations?
Use combinations when order doesn't matter — choosing a committee, selecting lottery numbers, dealing cards. Use permutations when order matters — arranging books on a shelf, setting a PIN code, ranking contestants. Memory trick: Combination = Choose (C for Choose); Permutation = Position (P for Position). Example: 5 people, choose 3: 10 combinations but 60 permutations (6× more because each combo has 3!=6 arrangements).
What does conditional probability mean in real life?
Conditional probability P(A|B) asks: given that B happened, how does that change the probability of A? Medical testing is a classic example: P(disease | positive test) depends on both test accuracy and disease prevalence (base rate). The famous "base rate fallacy" occurs when people ignore the base rate. A test that's 99% accurate doesn't mean a positive result means 99% chance of disease — you must account for how common the disease is.