Probability Calculator
Find the chance of both, either, or neither of two events.
What it does
Probability places a number between 0 and 1 on how likely an event is. Two events can be independent, dependent, or mutually exclusive. The label matters because the same two probabilities can produce different answers for both events and either event.
This calculator combines two events. It reports the chance of both, either, neither, and only A. For dependent events, it also uses P(B given A), the chance that B happens when A is known to have happened.
The core rules are short. Independent events multiply for the intersection. The union adds both probabilities and subtracts the overlap so it is not counted twice. Mutually exclusive events have no overlap, so their intersection is zero.
The tool checks that the inputs form a valid probability model, but it cannot know whether the relationship you chose is correct. Independence and exclusivity are assumptions about the situation. It also handles only two events, not a chain of conditional events.
How to use
- Choose independent, mutually exclusive, or dependent.
- Enter P(A) as a decimal between 0 and 1.
- Enter P(B) as a decimal between 0 and 1.
- For dependent events, enter P(B given A).
- Read P(A and B), P(A or B), P(neither), and P(A only).
- Check that the four displayed values form a consistent picture of the two events.
Formula
- P(A)
- Probability that event A occurs.
- P(B)
- Probability that event B occurs.
- P(B|A)
- Probability of B given that A occurred.
- P(A and B)
- Probability that both events occur.
- P(A or B)
- Probability that at least one event occurs.
For mutually exclusive events, P(A) + P(B) cannot exceed 1. For dependent events, P(A) x P(B|A) cannot exceed P(B), otherwise the conditional probability is inconsistent with the stated P(B). Percentages are rounded to one decimal place after the calculation.
Worked example
Example: independent events with P(A)=0.5 and P(B)=0.3
Relationship independent, P(A) 0.5, P(B) 0.3.
- Both: 0.5 x 0.3 = 0.15, or 15.0 %.
- Either: 0.5 + 0.3 - 0.15 = 0.65, or 65.0 %.
- Neither: 1 - 0.65 = 0.35, or 35.0 %.
- Only A: 0.5 - 0.15 = 0.35, or 35.0 %.
- Check the full split: only A 35.0 %, only B 15.0 %, both 15.0 %, neither 35.0 %, which totals 100.0 %.
- The tool shows 15.0 %, 65.0 %, 35.0 %, and 35.0 %.
Reference table
| Relationship | P(A) | P(B) | P(A and B) | P(A or B) |
|---|---|---|---|---|
| Independent | 0.50 | 0.30 | 15.0 % | 65.0 % |
| Independent | 0.25 | 0.40 | 10.0 % | 55.0 % |
| Dependent, P(B|A)=0.6 | 0.50 | 0.30 | 30.0 % | 50.0 % |
| Mutually exclusive | 0.50 | 0.30 | 0.0 % | 80.0 % |
Limits
- The calculator handles exactly two events, not a larger network.
- The relationship must be known from the situation, not inferred from the numbers.
- All inputs must be valid probabilities between 0 and 1.
- Independence is an assumption, not something this tool proves.
Common mistakes
- Adding P(A) and P(B) without subtracting the overlap.
- Multiplying probabilities for mutually exclusive events.
- Using P(B) instead of P(B|A) when events are dependent.
- Entering 30 instead of 0.30 for a 30% chance.
- Assuming two events are independent just because they look unrelated.
Try it yourself
Quick checks. Answers sit tucked under each question.
Independent events with P(A)=0.25 and P(B)=0.40: what are both, either, and neither?
Both 10.0 %, either 55.0 %, neither 45.0 %.
P(A)=0.25, P(B)=0.40, and P(B|A)=0.50: what are both and either?
Both 12.5 %, either 52.5 %, neither 47.5 %.
Mutually exclusive events with P(A)=0.20 and P(B)=0.30: what are both and either?
Both 0.0 %, either 50.0 %, neither 50.0 %.
Questions
What does independent mean in probability?
Knowing that one event happened does not change the chance of the other. In symbols, P(B|A) equals P(B).
What does mutually exclusive mean?
The two events cannot both happen in the same trial, so their intersection is zero.
Can P(A or B) exceed 1?
No. A valid probability model keeps the union at or below 1. The tool returns an error if the inputs produce a larger value.
What is P(A only)?
The chance that A happens and B does not. It equals P(A) minus the chance that both happen.
How do I find P(B given A)?
Estimate it from data or a model for the situation. You cannot derive it from P(B) unless the events are independent.
Can I enter percentages instead of decimals?
No. Enter 30% as 0.30. The tool treats numbers above 1 as invalid probabilities.