Probability Calculator
Calculate event probability, combinations, and permutations
Probability Calculator
Calculate event probabilities and combinations
Find the probability of an event and its complement
P(A) = favorable outcomes / total outcomesWhat is a Probability Calculator?
A Probability Calculator is a math tool that helps you find the likelihood of an event happening without doing the calculations manually. Probability measures chance as a number between 0 and 1, where 0 means an event is impossible and 1 means it is certain. Probabilities are also commonly written as percentages (0% to 100%) or as fractions.
Probability is used in many real-world situations: predicting outcomes in games (coins, dice, cards), analyzing risk in finance and insurance, estimating outcomes in science and medicine, and making decisions under uncertainty. Even simple probability skills can help you interpret data and understand everyday "odds."
This calculator is useful for quickly computing probabilities for common scenarios—like "favorable outcomes vs total outcomes"—and, depending on the calculator features, it may also help with combined events (AND/OR), complements (NOT), or conditional probability.
How to Use This Probability Calculator
- Choose the probability type -- such as single-event probability, two events (AND / OR), or conditional probability
- Enter the required values -- such as favorable outcomes (successes), total outcomes (all possible outcomes), or probabilities for Event A and Event B
- Click "Calculate" -- to get the probability
- Review the result -- in decimal, fraction, or percent (depending on what the calculator displays)
- Double-check assumptions -- equally likely outcomes, independence, and whether events overlap
Tips:
- Make sure total outcomes is greater than zero
- If you're using counts (favorable/total), the outcomes should be based on the same sample space
- For multi-event probability, confirm whether events are independent (one does not affect the other) or dependent (one affects the other)
Probability Formulas
Basic Probability (Equally Likely Outcomes)
P(A) = Favorable outcomes / Total outcomes
The ratio of successful outcomes to all possible outcomes
Complement (NOT A)
P(not A) = 1 − P(A)
The probability of an event NOT happening
Conditional Probability
P(B|A) = P(A ∩ B) / P(A)
Probability of B given that A has occurred
Addition Rule (A OR B)
Mutually exclusive events
P(A ∪ B) = P(A) + P(B)
Cannot happen at the same time
Overlapping events
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Subtract the overlap to avoid double-counting
Multiplication Rule (A AND B)
Independent events
P(A ∩ B) = P(A) × P(B)
One does not affect the other
Dependent events
P(A ∩ B) = P(A) × P(B|A)
One event affects the other
Example Calculations
Example 1: Coin Flip (Heads)
Setup: A fair coin has 2 equally likely outcomes
Favorable outcomes: 1 (Heads)
Total outcomes: 2
Calculation: P(Heads) = 1/2 = 0.5 = 50%
Result: 50%
Example 2: Rolling a 4 on a Die
Setup: A standard die has 6 outcomes (1–6)
Favorable outcomes: 1 (rolling a 4)
Total outcomes: 6
Calculation: P(4) = 1/6 ≈ 0.1667 = 16.67%
Result: 16.67%
Example 3: Rolling an Even Number
Even numbers on a die: 2, 4, 6
Favorable outcomes: 3
Total outcomes: 6
Calculation: P(Even) = 3/6 = 1/2 = 50%
Result: 50%
Example 4: Two Independent Events (AND)
Problem: Rolling a 6 AND flipping Heads
P(6): 1/6, P(Heads): 1/2
Calculation: P(6 AND Heads) = 1/6 × 1/2 = 1/12 ≈ 0.0833
Result: 8.33%
Example 5: A OR B with Overlap
Setup: Pick a random number from 1 to 10
A: "number is even" (2,4,6,8,10) → P(A) = 5/10
B: "number > 6" (7,8,9,10) → P(B) = 4/10
Overlap: A ∩ B = (8,10) → P(A ∩ B) = 2/10
Calculation: P(A ∪ B) = 5/10 + 4/10 − 2/10 = 7/10 = 70%
Result: 70%
Frequently Asked Questions
What does probability mean in simple terms?
Probability is the chance that something happens. It ranges from 0 (impossible) to 1 (certain), and it's often shown as a percentage from 0% to 100%.
What are "favorable outcomes" and "total outcomes"?
Favorable outcomes are the results you want (successes). Total outcomes are all possible results. For a die, total outcomes are 6; favorable outcomes depend on your event.
What's the difference between independent and dependent events?
Independent events do not affect each other (coin flip + die roll). Dependent events do affect each other (drawing two cards without replacement changes the second probability).
When can I add probabilities, and when should I not?
You can add probabilities when calculating A OR B, but if events overlap, you must subtract the overlap:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If events are mutually exclusive, the overlap is zero.
Why might a probability calculator result look "wrong"?
Common reasons include using the wrong sample space, assuming outcomes are equally likely when they aren't, mixing dependent and independent event rules, or forgetting overlap in OR problems.
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What Is Probability?
Probability is a measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It's fundamental to statistics, science, gambling, insurance, and machine learning. A probability of 0.5 means an event is equally likely to happen or not — like flipping a fair coin. Understanding probability helps you make better decisions under uncertainty in virtually every field.
This calculator handles single events, combined events (AND/OR), complements, conditional probability, combinations (choosing r items from n without regard to order), and permutations (ordered arrangements). Whether you're solving a statistics homework problem, designing an experiment, or just satisfying your curiosity, every calculation includes a clear result in both decimal and percentage form.
How to Use the Probability Calculator
- Select the type of calculation — single event, combined events (AND/OR), complement, conditional probability, or combinations/permutations.
- Enter the number of favorable outcomes and total outcomes (or n and r for combinations and permutations).
- Click Calculate.
- Read the probability as a decimal and percentage — results update instantly.
Probability Formulas
Single event: P(A) = favorable / total
Complement: P(A') = 1 − P(A)
AND (independent): P(A∩B) = P(A) × P(B)
OR: P(A∪B) = P(A) + P(B) − P(A∩B)
Conditional: P(A|B) = P(A∩B) / P(B)
Combinations: C(n,r) = n! / (r!(n−r)!)
Permutations: P(n,r) = n! / (n−r)!Probabilities always sum to 1 for all mutually exclusive outcomes. For independent events, AND means multiply; OR means add then subtract the overlap to avoid counting it twice. Combinations ignore order; permutations count every distinct ordering as a separate outcome.
Worked Examples
Rolling a 6 on a standard die
There is 1 favorable outcome (the face showing 6) out of 6 total outcomes. P = 1/6 ≈ 0.1667, or about 16.67%. The complement — not rolling a 6 — is P(A') = 1 − 1/6 = 5/6 ≈ 0.8333.
Drawing a heart from a standard deck of cards
A standard deck has 52 cards, 13 of which are hearts. P = 13/52 = 0.25, or exactly 25%. If you draw two cards without replacing the first, the events are dependent and the conditional formula applies.
Choosing 2 items from 5 — how many combinations?
C(5, 2) = 5! / (2! × 3!) = (5 × 4) / (2 × 1) = 10. There are 10 ways to choose 2 items from a group of 5 when order doesn't matter. If order matters, P(5, 2) = 5! / 3! = 20 permutations.