Probability Calculator
Calculate single and multiple event probabilities with step-by-step solutions.
Outcomes that count as success.
All possible outcomes in the sample space.
Probability result
30%
Basic probability
Complement
70%
Probability the event does not happen.
Decimal form
0.3
Fraction form
3/10
Odds in favor
3:7
Mode
Basic probability
Formula used
P(Event) = Favorable Outcomes ÷ Total Outcomes
3 ÷ 10
Probability Formulas and Rules
Basic Probability
P(Event) = Favorable Outcomes ÷ Total Outcomes
Complement
P(not A) = 1 − P(A)
Independent Events
P(A and B) = P(A) × P(B)
Union Rule
P(A or B) = P(A) + P(B) − P(A and B)
Conditional Probability
P(A | B) = P(A and B) ÷ P(B), when P(B) > 0
Mutually Exclusive Events
P(A or B) = P(A) + P(B)
Odds from Probability
Odds in favor = P(Event) ÷ P(not Event)
Variable Explanations
P(A)
Probability of event A.
P(B)
Probability of event B.
P(A and B)
Probability both events happen.
P(A or B)
Probability at least one event happens.
P(A | B)
Probability of A given B.
Complement
Probability an event does not happen.
Favorable outcomes
Outcomes that count as success.
Total outcomes
All possible outcomes in the sample space.
What Probability Means
Likelihood
Probability measures how likely an event is.
Range
Probability ranges from 0 to 1.
Impossible
0 means the event cannot happen.
Certain
1 means the event must happen.
Equally likely
0.5 means the event and its complement are equally likely.
Multiple forms
Probability can be written as a decimal, fraction, or percentage.
Worked Examples
Rolling a 6 on a fair die
Formula: P = favorable ÷ total
Substitution: 1 ÷ 6
Answer: 0.1667 = 16.67%
One face out of six is a 6.
Drawing a red card
Formula: P = favorable ÷ total
Substitution: 26 ÷ 52
Answer: 0.5 = 50%
Half the cards in a standard deck are red.
Complement probability
Formula: P(not A) = 1 − P(A)
Substitution: 1 − 0.30
Answer: 0.70 = 70%
If A has a 30% chance, not A has a 70% chance.
Coin flip and die roll
Formula: P(A and B) = P(A) × P(B)
Substitution: 0.5 × 1/6
Answer: 0.0833 = 8.33%
Independent events multiply.
Union probability
Formula: P(A or B) = P(A) + P(B) − P(A and B)
Substitution: 0.4 + 0.3 − 0.1
Answer: 0.6 = 60%
Subtract overlap once.
Conditional probability
Formula: P(A | B) = P(A and B) ÷ P(B)
Substitution: 0.12 ÷ 0.30
Answer: 0.4 = 40%
The sample space is limited to B.
Odds from probability
Formula: Odds = P ÷ (1 − P)
Substitution: 0.25 ÷ 0.75
Answer: 1:3
One success for every three failures.
Invalid setup
Formula: Favorable cannot exceed total
Substitution: 12 ÷ 10
Answer: Invalid
Favorable outcomes cannot be greater than total outcomes.
Independent, Dependent, and Conditional Probability
Independent
Events do not affect each other. Example: flipping a coin twice.
Dependent
Events affect each other. Example: drawing cards without replacement.
Conditional
Probability after new information is known. Example: P(A given B).
Probability Rules and Complements
Complement rule
P(not A) = 1 − P(A).
Union rule
Add probabilities and subtract overlap.
Intersection rule
For independent events, multiply probabilities.
Mutually exclusive
Events cannot happen at the same time.
Exhaustive outcomes
All possible outcomes together have probability 1.
Overlap
Overlapping events require subtracting the overlap once.
Common Use Cases
Common Probability Mistakes
Understanding Your Results
Probability result
Likelihood of the event.
Percent form
Probability out of 100.
Decimal form
Probability from 0 to 1.
Fraction form
Favorable share of the whole.
Complement
Probability the event does not happen.
Odds
Comparison of happening vs not happening.
Frequently Asked Questions
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