Probability Calculator
Calculate probability from favorable and total equally likely outcomes, convert the result to odds and percentage, combine two independent events with AND or OR, find the probability of neither event, or calculate a complement.
Probability result
P(A): 0.3 (30%)
Odds in favor: 3:7
Odds against: 7:3
How to Use the Probability Calculator
Choose Single event to calculate probability from favorable and total equally likely outcomes. Choose Two independent events to combine P(A) and P(B), or choose Complement to find the probability that an event does not occur.
Single-event mode shows the result as a decimal and percentage together with odds in favor and odds against.
For the two-event and complement modes, enter probabilities as decimals from 0 to 1. For example, enter 0.25 for 25% and 0.8 for 80%.
What Is Probability?
Probability measures how likely an event is to occur. A probability of 0 represents an impossible event, while a probability of 1 represents certainty.
Probabilities between 0 and 1 describe intermediate likelihoods. A probability of 0.5 is the same as 50%, while 0.25 is 25%.
Probability can be written as a decimal, fraction, or percentage depending on the problem.
| Probability | Percentage | Interpretation |
|---|---|---|
| 0 | 0% | Impossible event |
| 0.1 | 10% | 1 chance in 10 |
| 0.25 | 25% | 1 chance in 4 |
| 0.5 | 50% | Equal chance either way |
| 0.75 | 75% | 3 chances in 4 |
| 0.9 | 90% | 9 chances in 10 |
| 1 | 100% | Certain event |
Probability From Favorable Outcomes
When all possible outcomes are equally likely, the probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.
For example, if 3 of 10 equally likely outcomes count as success, the probability is 3/10 = 0.3 = 30%.
This counting method depends on the outcomes being equally likely. Simply counting outcomes is not enough when some possibilities are more likely than others.
Probability as a Decimal, Fraction, and Percentage
The same probability can be represented in several forms. For example, 1/4, 0.25, and 25% all describe the same likelihood.
To convert a decimal probability to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.
This calculator displays percentages alongside its decimal probability results, but probability inputs in the independent-event and complement modes are entered as decimals.
Probability vs Odds
Probability compares favorable outcomes with all possible outcomes. Odds compare favorable outcomes directly with unfavorable outcomes.
If an event has 3 favorable outcomes and 7 unfavorable outcomes, its probability is 3/10 = 30%, while its odds in favor are 3:7.
Odds against reverse the comparison. In the same example, the odds against the event are 7:3.
| Favorable | Unfavorable | Probability | Odds in favor | Odds against |
|---|---|---|---|---|
| 1 | 1 | 50% | 1:1 | 1:1 |
| 1 | 3 | 25% | 1:3 | 3:1 |
| 3 | 1 | 75% | 3:1 | 1:3 |
| 2 | 3 | 40% | 2:3 | 3:2 |
Odds in Favor and Odds Against
Odds in favor are written as favorable : unfavorable. Odds against are written as unfavorable : favorable.
For an event with probability 75%, imagine 3 favorable outcomes and 1 unfavorable outcome. The odds in favor are 3:1 and the odds against are 1:3.
A probability of 0 produces odds in favor of 0:1, while a certain event produces odds in favor of 1:0.
Two Independent Events
Two events are independent when knowing whether one occurred does not change the probability of the other.
Examples can include separate fair coin flips or repeated rolls of a fair die when each trial does not affect the next one.
The two-event mode in this calculator assumes independence. If one event changes the probability of the other, these formulas should not be used without conditional probabilities.
Probability of A AND B
A AND B means both events occur.
For independent events, multiply their probabilities: P(A and B) = P(A) × P(B).
For example, if P(A) = 0.3 and P(B) = 0.5, the probability that both occur is 0.3 × 0.5 = 0.15, or 15%.
Probability of A OR B
A OR B means A occurs, B occurs, or both occur.
The general addition rule is P(A or B) = P(A) + P(B) - P(A and B). The overlap is subtracted so outcomes where both occur are not counted twice.
For independent events, P(A and B) equals P(A)P(B), so this calculator uses P(A) + P(B) - P(A)P(B).
Probability of Neither Event
Neither A nor B means that A does not occur and B does not occur.
For independent events, the probability is (1 - P(A))(1 - P(B)).
If P(A) = 0.3 and P(B) = 0.5, then the probability of neither is 0.7 × 0.5 = 0.35, or 35%.
At Least One Event Occurs
For two events, at least one occurs means A occurs, B occurs, or both occur. This is the same event represented by A OR B.
Another way to calculate it is to find the complement of neither: P(at least one) = 1 - P(neither).
For independent events with probabilities 0.3 and 0.5, neither has probability 0.35, so at least one occurs with probability 0.65.
Complement Probability
The complement of an event A is the event that A does not occur.
Because either A occurs or it does not, their probabilities add to 1. Therefore P(not A) = 1 - P(A).
If an event has probability 0.72, its complement has probability 0.28, or 28%.
Independent vs Mutually Exclusive Events
Independent and mutually exclusive do not mean the same thing.
Independent events can occur together, and the occurrence of one does not change the probability of the other. Mutually exclusive events cannot occur together at all.
For two events with positive probability, being mutually exclusive generally means they are not independent because learning that one occurred tells you the other did not occur.
When the Independence Formula Does Not Apply
For dependent events, P(A and B) is not generally equal to P(A)P(B). Instead, conditional probability must be considered.
For example, drawing two cards from a deck without replacement creates dependent events because the first draw changes the composition of the deck.
The independent-event mode should therefore be used only when P(A) remains the same regardless of whether B occurs, and vice versa.
Impossible and Certain Events
A probability of 0 means the event cannot occur under the stated model. A probability of 1 means it is certain.
These endpoint probabilities are valid inputs in this calculator.
In single-event mode, zero favorable outcomes produces probability 0, while favorable outcomes equal to total outcomes produces probability 1.
Common Probability Mistakes
Counting outcomes that are not equally likely: Favorable divided by total is valid only when the counted elementary outcomes have equal probability.
Adding probabilities for AND: Independent events occurring together use multiplication, not simple addition.
Forgetting overlap in OR: P(A) + P(B) counts the intersection twice unless the events are mutually exclusive.
Confusing independence with mutual exclusivity: Independent events may occur together; mutually exclusive events cannot.
Entering percentages as whole numbers: In decimal-input modes, enter 25% as 0.25, not 25.
Confusing probability with odds: A probability of 25% corresponds to odds in favor of 1:3, not 25:75 written as a percentage.
How This Probability Calculator Works
Single-event mode divides favorable outcomes by total outcomes and compares favorable outcomes with unfavorable outcomes to produce simplified odds.
Independent-event mode multiplies P(A) and P(B) to find the intersection, then applies the addition rule to find P(A or B). It also multiplies the two complements to find the probability of neither event.
Complement mode subtracts the entered probability from 1.
All probability calculations use the numerical values entered and do not estimate probabilities from experimental data.
Probability Formulas
The two-event AND, OR, and neither formulas shown here use the independence assumption built into this calculator.
- Probability of event A
- Probability of event B
- A and B both occur
- A or B or both occur
- Complement of A; A does not occur
Examples
Three favorable outcomes
3 favorable outcomes out of 10 equally likely outcomes
P(A) = 0.3 = 30%; odds in favor 3:7; odds against 7:3.
Coin flip
1 favorable outcome out of 2 equally likely outcomes
P(A) = 0.5 = 50%; odds in favor 1:1.
Rolling a six
1 favorable result out of 6 equally likely die faces
P(A) ≈ 0.166667 = 16.6667%; odds in favor 1:5.
Independent AND
P(A) = 0.3; P(B) = 0.5
P(A and B) = 0.15 = 15%.
Independent OR
P(A) = 0.3; P(B) = 0.5
P(A or B) = 0.65 = 65%.
Neither independent event
P(A) = 0.3; P(B) = 0.5
P(neither) = 0.35 = 35%.
Complement
P(A) = 0.72
P(not A) = 0.28 = 28%.
Certain event
10 favorable outcomes out of 10
Probability 1 = 100%; odds in favor 1:0.
Frequently Asked Questions
How do I calculate probability?
For equally likely outcomes, divide the number of favorable outcomes by the total number of possible outcomes.
How do I convert probability to a percentage?
Multiply the decimal probability by 100. A probability of 0.35 is 35%.
Can I enter percentages directly?
Not in the decimal-probability fields. Enter 30% as 0.3, 5% as 0.05, and 100% as 1.
What is the difference between probability and odds?
Probability compares favorable outcomes with all outcomes. Odds in favor compare favorable outcomes with unfavorable outcomes.
What are the odds for a 25% probability?
A probability of 25% can be represented as 1 favorable outcome and 3 unfavorable outcomes, so the odds in favor are 1:3.
What does AND mean in probability?
A AND B means that both events occur. For independent events, multiply P(A) by P(B).
What does OR mean in probability?
A OR B means A occurs, B occurs, or both occur.
Why do I subtract the overlap for OR?
Adding P(A) and P(B) counts outcomes where both events occur twice, so their intersection must be subtracted once.
What is the probability that neither event occurs?
For independent events, multiply their complements: (1 - P(A))(1 - P(B)).
How do I find the probability of at least one event?
For two events, at least one is the same as A OR B. You can also subtract the probability of neither event from 1.
What is a complement in probability?
The complement of A means A does not occur. Its probability is 1 - P(A).
What are independent events?
Events are independent when the occurrence of one does not change the probability of the other.
Can I use this for dependent events?
Single-event and complement calculations are fine when applicable, but the two-event mode assumes independence and is not designed for dependent-event conditional probabilities.
Are independent and mutually exclusive events the same?
No. Independent events can occur together without affecting each other's probability. Mutually exclusive events cannot occur together.
Can probability be greater than 1?
No. A probability must be between 0 and 1 inclusive, equivalent to 0% through 100%.
What does probability 0 mean?
It means the event is impossible under the probability model being used.
What does probability 1 mean?
It means the event is certain under the probability model being used.
Why must single-event outcomes be equally likely?
The simple favorable-divided-by-total formula treats each counted outcome as having the same probability. If outcomes have different probabilities, counting them alone does not determine the event probability.
What happens when favorable outcomes are zero?
The probability is 0 and the odds in favor are shown as 0:1.
What happens when every outcome is favorable?
The probability is 1, or 100%, and the odds in favor are shown as 1:0.