How to Use the Probability Calculator
- Choose the probability type that matches your problem.
- Enter outcome counts or probability values as requested.
- Press Calculate.
- Review the decimal, percentage, formula, interpretation, and step-by-step solution.
For counting possible arrangements before calculating probability, use the Permutation & Combination Calculator. For probabilities based on a bell curve, use the Normal Distribution Calculator.
Probability Formulas
Single event: P(A) = n(A) ÷ n(S)
Complement: P(A′) = 1 − P(A)
Independent AND: P(A ∩ B) = P(A) × P(B)
OR / union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B)
At least once in n independent trials: 1 − (1 − p)n
Probability measures how likely an event is to occur. It ranges from 0 for an impossible event to 1 for a certain event and may also be written as a fraction or percentage.
Single Event Probability
When all outcomes are equally likely, divide the number of favorable outcomes by the total number of possible outcomes. The complement gives the probability that the event does not occur.
Example: Rolling a Die
What is the probability of rolling a 6 on a fair six-sided die?
P(6) = 1 ÷ 6 = 0.1667
Answer: approximately 16.67%. The probability of not rolling a 6 is approximately 83.33%.
Probability of Two Events
AND / Intersection
For independent events, multiply the probabilities:
P(A ∩ B) = P(A)P(B)
For two independent fair coin flips, the probability of heads on both flips is:
0.5 × 0.5 = 0.25
Answer: 25%.
OR / Union
If events can overlap, add their probabilities and subtract the intersection once:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive, P(A ∩ B) = 0.
Conditional Probability
Conditional probability measures how likely A is when B is known to have occurred:
P(A | B) = P(A ∩ B) ÷ P(B)
Repeated Independent Trials
When the same independent trial is repeated n times with success probability p, the complement rule makes “at least once” problems especially simple:
If an event has a 20% chance per attempt and you try 3 independent times:
P(at least once) = 1 − (1 − 0.2)3
= 1 − 0.83 = 0.488
Answer: 48.8%.
Independent vs Dependent Events
Independent events: the occurrence of one event does not change the probability of the other. For example, separate fair coin flips are independent.
Dependent events: knowing that one event occurred changes the probability of another. Drawing cards without replacement is a common example because the sample space changes after the first draw.
Common Probability Mistakes
- Using invalid outcome counts: favorable outcomes cannot be negative or exceed the total number of outcomes.
- Assuming events are independent: if one event changes another event's probability, use conditional probability instead of automatically multiplying.
- Adding overlapping probabilities incorrectly: subtract P(A ∩ B) once when calculating P(A ∪ B).
- Confusing probability with odds: both describe likelihood, but they use different ratios.
How to Verify the Answer
A valid probability must lie between 0 and 1. For any event, P(A) + P(A′) = 1. For independent events, P(A ∩ B) should equal P(A)P(B). For OR calculations, the result must remain between 0 and 1 and should not be smaller than either individual event probability when the inputs are valid.
Probability Formula Reference
| Problem Type | Formula | Use When |
|---|---|---|
| Single event | P(A) = n(A) / n(S) | Outcomes are equally likely. |
| Complement | P(A′) = 1 − P(A) | You want the probability the event does not happen. |
| Independent AND | P(A ∩ B) = P(A)P(B) | A and B are independent and both must occur. |
| OR / union | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | At least one of A or B occurs. |
| Conditional | P(A | B) = P(A ∩ B) / P(B) | B is known to have occurred. |
| At least once | 1 − (1 − p)n | The same independent trial is repeated n times. |
Calculation Methodology
Utilixea applies standard probability rules based on the selected event type. Outcome-count inputs are validated as non-negative whole numbers, probability inputs are limited to the 0–1 range after conversion, and each result displays the formula used.
Last reviewed: July 22, 2026
Frequently Asked Questions
For equally likely outcomes, probability is favorable outcomes divided by total possible outcomes: P(A) = n(A) / n(S).
Use the complement rule: P(A′) = 1 − P(A).
If A and B are independent, multiply their probabilities: P(A ∩ B) = P(A)P(B).
For events that may overlap, use P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If the events are mutually exclusive, the intersection is 0.
Mutually exclusive events cannot occur at the same time, so P(A ∩ B) = 0.
Conditional probability is the probability of event A given that event B has occurred: P(A | B) = P(A ∩ B) / P(B), provided P(B) is greater than 0.
For n independent trials with success probability p, use 1 − (1 − p)n.
