OR Probability Calculator - Union of Two Events A and B
Use this free or probability calculator to find the probability that at least one of two events A or B occurs from P(A), P(B), and the joint probability P(A and B).
OR Probability Calculator
Results
What Is an OR Probability Calculator?
An or probability calculator is a probability-theory tool that turns the probability of event A, the probability of event B, and the probability that both events occur into the probability that at least one of A or B occurs. Type P(A) and P(B) as decimals between 0 and 1, type the joint P(A and B), and the result panel shows the union as a percent and a decimal, the complement, and the probability of exactly one.
- • Introductory probability homework: AP statistics students can plug in P(A), P(B), and the joint and read the union, complement, and exactly-one values.
- • Dice, card, and coin problems: Readers on dice rolls, card draws, or coin toss problems can map each event to a probability and read the at-least-one result.
- • Risk and reliability analysis: Engineers can combine failure probabilities and read the union of two failure modes and the probability that neither fails.
- • Survey and screening questions: Researchers combining two yes/no questions can read the share expected to answer yes to at least one.
The union rule P(A or B) = P(A) + P(B) - P(A and B) is the workhorse of set-arithmetic probability. The subtraction removes the double counting of the joint region. The same three probabilities drive the complement, the exactly-one split, and the per-event only probabilities.
Many textbook problems describe the events as independent or mutually exclusive, so the joint is either P(A) * P(B) or 0. The shortcut selector handles both cases.
Readers who need the complement, conditional, or single-event probabilities for the same A and B can move to the Probability Calculator and reuse P(A) and P(B) there.
How the OR Probability Calculator Works
The tool reads P(A), P(B), and the joint P(A and B), applies the shortcut selector, validates the joint against the Frechet bounds, and returns the union, complement, exactly-one, and per-event probabilities.
- P(A): Probability that A occurs. Real number in [0, 1].
- P(B): Probability that B occurs. Real number in [0, 1].
- P(A and B): Joint probability. P(A) * P(B) for independent, 0 for mutually exclusive.
- P(A or B): Union probability. Sits in [max(P(A), P(B)), min(1, P(A)+P(B))].
- P(neither): Complement of the union; equals 1 - P(A or B).
- P(exactly one): Symmetric difference; equals P(A) + P(B) - 2 * P(A and B).
Worked example: at least one head in two coin tosses
P(A) = 0.5, P(B) = 0.5, P(A and B) = 0.25.
P(A or B) = 0.5 + 0.5 - 0.25 = 0.75. P(neither) = 0.25. P(exactly one) = 0.5.
P(A or B) = 0.75 (75 percent). P(neither) = 0.25, P(exactly one) = 0.5.
Three of four toss pairs (HH, HT, TH) contain at least one head; only TT contains none.
According to Wolfram MathWorld (Inclusion-Exclusion Principle), the inclusion-exclusion principle gives the probability of the union as P(A) + P(B) - P(A and B), and for mutually exclusive events it collapses to P(A) + P(B).
Readers with a binomial n and p can pull the single-trial probability into this or probability calculator, then move the same p into the Binomial Distribution Calculator when the problem shifts from two trials to n.
Key Concepts Behind the OR Probability
Four short definitions keep the union formula honest.
Union of two events
The union A or B is the set of outcomes where at least one event occurs. Its probability equals P(A) plus P(B) minus the joint probability.
Joint probability P(A and B)
The joint probability is the probability that both events occur on the same trial. For independent events it equals P(A) * P(B); for mutually exclusive it equals 0.
Mutually exclusive events
Two events are mutually exclusive when they cannot occur together, so P(A and B) = 0 and the union simplifies to P(A) + P(B). Rolling a 1 or a 6 on the same die is the example.
Independent events
Two events are independent when the outcome of one does not change the probability of the other. The joint equals P(A) * P(B), so the union collapses to P(A) + P(B) - P(A) * P(B).
The Fréchet bounds govern every valid input. The joint probability P(A and B) cannot exceed the smaller single-event probability, so P(A and B) <= min(P(A), P(B)), and it cannot be smaller than P(A) + P(B) - 1 because the two events together cannot cover more than the entire sample space. The form rejects out-of-range values with an inline error above the result panel.
Readers who prefer fractions can keep the same A and B in the Probability Fraction Calculator and convert each fraction back to a decimal here.
How to Use the OR Probability Calculator
Type the three probability inputs and pick the shortcut that matches your problem.
- 1 Enter P(A): Type the probability that A occurs as a decimal in [0, 1]. For two coin tosses, P(head on toss 1) is 0.5; for a die, P(rolling a 1) is 1/6.
- 2 Enter P(B): Type the probability that B occurs the same way. P(head on toss 2) is 0.5; P(rolling a 6) is 1/6.
- 3 Enter P(A and B): Type the joint probability that both events occur.
- 4 Pick the shortcut: Choose Custom to keep the joint value, Independent events to auto-fill P(A) * P(B), or Mutually exclusive to set the joint to 0.
- 5 Read P(A or B): The first row shows P(A or B) as a percent; the second row shows the same value as a decimal.
- 6 Read the partition: The remaining rows show P(neither), P(exactly one), P(A only), P(B only), and the P(A) * P(B) reference. P(exactly one) equals P(A only) plus P(B only); the P(A) * P(B) row is just an independence reference.
A reader has two independent coin tosses and wants the probability of at least one head. They type 0.5 into P(A), 0.5 into P(B), choose Independent events, and read 75 percent as P(A or B), 0.25 as P(neither), and 0.50 as P(exactly one).
Readers who convert a Z-score into a tail probability can move the value through the Z-Score Calculator and into the P(A) box here.
Benefits of Using the OR Probability Calculator
A short list of what the tool does well, and where its job ends.
- • Three inputs, six outputs: Type P(A), P(B), and P(A and B) once and read the union, complement, exactly-one, and per-event only probabilities at the same time.
- • Independent and mutually exclusive shortcuts: The shortcut selector auto-fills P(A) * P(B) for independent events and 0 for mutually exclusive events.
- • Frechet bound validation: The form rejects joint probabilities outside [max(0, P(A)+P(B)-1), min(P(A), P(B))] so the result never reports an inconsistent set.
- • Percent and decimal together: The primary row shows the union as a percent, the second row shows the same value as a decimal.
- • Partition and reference rows: The result panel shows P(neither), P(A only), P(B only), P(exactly one), and the P(A) * P(B) reference; P(exactly one) equals P(A only) plus P(B only), and the P(A) * P(B) row is a comparison value, not a partition probability.
The form is built for the union-rule problems that appear in AP statistics and introductory probability. For conditioning or Bayes' theorem, the calculator is a stepping stone rather than a complete answer.
Readers moving from a two-trial union to a count-of-trials problem can route the same per-trial p into the Geometric Distribution Calculator for the first-success or first-failure distribution.
Factors That Affect Your OR Probability Result
The result depends on the three inputs and the relationship between the two events.
Magnitude of P(A) and P(B)
The union sits between max(P(A), P(B)) and min(1, P(A)+P(B)). Small probabilities give unions near P(A) + P(B); large ones make the subtraction matter more.
Joint probability P(A and B)
The joint subtracts from the simple sum, so doubling the joint from 0.1 to 0.2 lowers the union by 0.1.
Independent versus mutually exclusive
Independent events give P(A and B) = P(A) * P(B) and smaller unions; mutually exclusive events give 0 and the largest union P(A) + P(B).
Independence reference row
The P(A) * P(B) reference row shows what the joint would be if the events were independent; comparing it with the actual joint reveals how dependent the events are.
Frechet bounds
For any valid pair, P(A and B) must lie in [max(0, P(A)+P(B)-1), min(P(A), P(B))]. The form rejects out-of-range values and shows an inline error naming the violated range.
- • The union formula assumes P(A), P(B), and P(A and B) describe the same sample space and trial.
- • The shortcut selector assumes the events are independent or mutually exclusive. If the joint you typed is correct, keep Custom.
- • The form returns a point estimate with no confidence interval. Input probabilities carry uncertainty the union rule cannot propagate.
The P(A) * P(B) reference row is a useful sanity check. If the events are truly independent the joint you typed should match the reference; if the joint is much larger the events overlap heavily.
According to OpenStax Introductory Statistics 2e, 3.4 (LibreTexts), the addition rule states P(A or B) = P(A) + P(B) - P(A and B), and the multiplication rule for independent events states P(A and B) = P(A) * P(B).
Readers who model two independent rare-event rates as exponential waiting times can route the per-event rate into the Exponential Distribution Calculator for the inter-arrival distribution, and back into this or probability calculator for the at-least-one rate.
Frequently Asked Questions
Q: How do I find the probability of A or B?
A: Use the union rule P(A or B) = P(A) + P(B) - P(A and B). Type P(A), P(B), and the joint probability P(A and B) into the form and the result panel shows the union as a percent and as a decimal, plus the complement and the exactly-one split.
Q: What is the OR probability formula for independent events?
A: For independent events the joint probability equals P(A) * P(B), so the union formula becomes P(A) + P(B) - P(A) * P(B). The shortcut selector in the form auto-fills P(A) * P(B) into the joint field so the general formula still produces the correct answer.
Q: What is the OR probability formula for mutually exclusive events?
A: For mutually exclusive events P(A and B) equals 0, so the union simplifies to P(A) + P(B). Rolling a 1 or a 6 on the same die toss is the classic example: 1/6 + 1/6 = 1/3.
Q: How do I calculate the probability of at least one of two events?
A: Use the union rule P(A or B) = P(A) + P(B) - P(A and B). The result panel shows the same value as a percent and as a decimal, and the second row exposes the complement P(neither A nor B) so you can cross-check that the two values sum to 1.
Q: How do I represent the OR probability in a Venn diagram?
A: Draw two overlapping circles, label one A and one B, and color the union - both circles plus their overlap - as the OR region. The probability P(A or B) is the total colored area, while the joint region is the lens-shaped intersection, and the complement is the area outside both circles.
Q: What is the probability that neither of two events occurs?
A: It equals the complement of the union, so P(neither A nor B) = 1 - P(A or B). For two independent fair-coin tosses the union is 0.75 and the complement is 0.25, matching the single TT outcome out of four equally likely pairs.