Event probabilities and binomial success counts
The main result is a percentage with its complement and decimal equivalent. Single-event odds use favorable to unfavorable outcomes; binomial and two-event odds are approximate. Expected successes is n times the single-trial probability. At most and at least appear only for binomial trials and both include the selected success count. Open the breakdown for working and, in Two events mode, conditional chances.
If you have a list of observed outcomes to summarize, the Mean, Median & Mode Calculator describes their center and range rather than their theoretical likelihood.
The formula
Binomial probabilities count arrangements of k successes, with cumulative results adding the appropriate exact probabilities. For independent events, the joint probability is P(A) × P(B). For mutually exclusive events it is zero; otherwise supply the known overlap. A given B is the joint probability divided by P(B), and is undefined when P(B) is zero.
Two heads in four fair coin tosses
With four independent trials and a 50% success chance, exactly two successes has probability 6 × 0.5⁴ = 37.5%. At most two and at least two both equal 68.75%.
How to use this calculator
- Choose Equally likely outcomes, Binomial trials or Two events. Only the fields for that question remain visible.
- For outcome counts, enter favorable and total outcomes plus independent repetitions for the expectation. For binomial trials, enter the per-trial chance as a percentage, number of trials and selected successes.
- For two events, enter each percentage and choose Independent, Mutually exclusive or Known overlap. Supply their joint chance when using Known overlap, then select the probability to highlight.
- Answers update as inputs change. Calculate updates immediately; open Probability breakdown and working for all event combinations or cumulative calculations. Reset restores the initial example.
Binomial assumptions matter
A binomial model uses a fixed number of independent trials with only success or failure on each trial.
The success probability must stay constant. Sampling without replacement from a small group can violate independence and require a different model.
Expected value is a long-run average
Multiplying trials by the success probability gives the expected number of successes across repeated experiments.
An expected value need not be a possible outcome in one experiment; three trials at 50% have an expectation of 1.5 successes.
Two events need a relationship
Knowing each event’s chance does not determine the chance of both. Independence justifies multiplying probabilities; it should not be assumed for events that affect each other.
An overlap must be at least max(0, P(A) + P(B) − 1) and at most min(P(A), P(B)). These bounds ensure that every event combination has a valid probability. For example, independent events with chances of 60% and 75% have a 45% joint chance and a 90% chance of either or both.
Assumptions & limitations
What this calculation assumes
- Single outcomes are equally likely.
- Binomial trials are independent with a constant success probability.
- Cumulative probabilities include the selected success count.
- Two-event calculations use the selected independence, mutual-exclusion or known-overlap relationship.
What to keep in mind
- The binomial model does not cover changing probabilities or dependent trials.
- Results use floating-point approximations.
- Probability estimates do not guarantee an outcome.
- Conditional probabilities are undefined for a conditioning event with zero chance. The tool does not infer missing probabilities from arbitrary combinations of known values.
Common questions
What is the complement of an event?
It is the probability that the event does not occur, calculated as one minus the event probability.
What does at least k include?
It includes exactly k successes and every larger possible success count.
Can probability exceed 100%?
No. Valid probability lies from 0% to 100%, inclusive.
Why do at most k and at least k add to more than 100%?
Both include exactly k. In the four-toss example, 68.75% + 68.75% = 137.5%, which is 100% plus the 37.5% chance of exactly two heads.
Are independent and mutually exclusive events the same?
No. Independent events do not change one another’s chance. Mutually exclusive events cannot occur together; two such events with positive probabilities are not independent.
Does either event include both events?
Yes. Either or both is the inclusive union. Choose Exactly one to exclude the overlap.
Sources & further reading
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