Probability Calculator

About the Probability Calculator

Probability measures how likely an event is on a scale from 0 (impossible) to 1 (certain). Most textbook questions fall into a handful of patterns: counting favorable outcomes, combining independent events, conditioning on information you already have, and counting successes in repeated trials. This calculator handles all four.

Inputs can be whole-number counts, decimals such as 0.25, fractions such as 1/4 or percentages such as 25%. Wherever possible the answer is given as an exact fraction alongside its decimal and percentage, so you can match the form your teacher or textbook expects. Each mode also shows the complementary or related probabilities that questions usually ask for next.

What is probability?

When all outcomes of an experiment are equally likely, the probability of an event $A$ is the number of outcomes in $A$ divided by the total number of outcomes. Rolling two dice has 36 equally likely outcomes, and six of them sum to 7, so $P(\text{sum} = 7) = \frac{6}{36} = \frac16$ .

Probabilities obey a few rules: they lie between 0 and 1, the probability that $A$ does not happen is $1 - P(A)$ , and the probability that $A$ or $B$ happens is $P(A) + P(B) - P(A \cap B)$ . Conditional probability and the binomial distribution build directly on these rules.

How to use the Probability Calculator

Choose the problem type: single event, two independent events, conditional probability or binomial.
For a single event, enter the number of favorable outcomes and the total number of equally likely outcomes.
For independent events, enter P(A) and P(B); for conditional probability, enter P(A and B) and P(B).
For a binomial problem, enter the number of trials n, the success probability p, the number of successes k and whether you want exactly, at most, fewer than, at least or more than k.
Press Calculate probability and read the exact answer, the percentage and the supporting steps.

Formulas

Classical probability

P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

Independent events

P(A \cap B) = P(A)\,P(B)

Conditional probability

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Binomial probability

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Worked example: Two dice: P(sum 7) = 6/36

The example asks for the probability that two fair dice sum to 7. There are 6 favorable outcomes, (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), out of 36, so $P = \frac{6}{36} = \frac16 \approx 0.1667$ , or about 16.67%.

The calculator also gives the complement $1 - \frac16 = \frac56$ , the probability of any other total, and the odds in favor, $6 : 30$ , which compare favorable outcomes with unfavorable ones rather than with the total.

The solution below is generated by the same engine that powers the calculator above.

Final answer

P(A) = \frac{1}{6} \approx 0.16666667

How we read your problem

Single event (favorable ÷ total)

Step-by-step solution

Divide favorable outcomes by total equally likely outcomes
$P(A) = \frac{6}{36} = \frac{1}{6} \approx 0.16666667$
Complement: the event does not happen
$P(A') = 1 - P(A) = \frac{5}{6} \approx 0.83333333$
Odds in favor
$6 : 30$

Formulas used

Classical probability
$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$
Complement
$P(A') = 1 - P(A)$

The event occurs in 6 of 36 equally likely outcomes, so its probability is 16.666667%; it fails with probability 83.333333%.

Detailed explanation

Independent events and the addition rule

Events are independent when knowing that one happened does not change the probability of the other, like separate coin flips. Then the probability of both is the product. With $P(A) = \frac12$ and $P(B) = 0.3$ , both happen with probability $0.15$ , at least one with $0.5 + 0.3 - 0.15 = 0.65$ , exactly one with $0.65 - 0.15 = 0.5$ and neither with $0.5 \times 0.7 = 0.35$ .

The calculator checks that $P(A \cup B) + P(\text{neither}) = 1$ , a useful sanity test because together those two cases cover every outcome.

Conditional probability

Conditional probability narrows the sample space to the outcomes where $B$ happened, then asks how often $A$ also happens inside that smaller space. If 12% of customers buy both a phone and a case and 40% buy a phone, then $P(\text{case} \mid \text{phone}) = \frac{0.12}{0.4} = 0.3$ , so 30% of phone buyers also take a case.

The joint probability can never exceed $P(B)$ , and $P(B)$ must be positive; the calculator reports an error in either situation. Remember that $P(A \mid B)$ and $P(B \mid A)$ are usually different, and Bayes' theorem is the tool that converts one into the other.

Binomial probability

A binomial experiment repeats $n$ independent trials, each with the same success probability $p$ . The probability of exactly $k$ successes multiplies the number of ways to place them, $\binom{n}{k}$ , by the probability of any one such arrangement. For at most, at least, fewer than or more than $k$ , the calculator adds the relevant terms and states which ones it summed.

For example, the probability of at least 8 heads in 10 fair coin flips is $\frac{45 + 10 + 1}{1024} \approx 0.0547$ . The calculator also reports the mean $np$ and standard deviation $\sqrt{np(1-p)}$ of the distribution.

Common mistakes

Multiplying probabilities of events that are not independent.
Adding $P(A)$ and $P(B)$ without subtracting the overlap, which can give a total above 1.
Confusing $P(A \mid B)$ with $P(B \mid A)$ .
Counting outcomes that are not equally likely, such as treating the sums 2 to 12 of two dice as equally probable.
Mixing up odds and probability; odds of 1 : 5 correspond to a probability of $\frac16$ .

Tips

Use the complement for 'at least one' questions: $1 - P(\text{none})$ is usually quicker.
List or draw the sample space for small problems before computing.
Enter probabilities as fractions to keep the answer exact.
Check that every answer lies between 0 and 1.
Use the permutation and combination calculator to count outcomes before dividing.

Frequently asked questions

How do I calculate the probability of an event?

Divide the number of favorable outcomes by the total number of equally likely outcomes. For a sum of 7 with two dice that is 6 ÷ 36 = 1/6.

How do I find the probability of two independent events both happening?

Multiply their probabilities. With P(A) = 0.5 and P(B) = 0.3, both happen with probability 0.15.

What is conditional probability?

The probability of A given that B has happened, P(A | B) = P(A and B) ÷ P(B). It restricts attention to outcomes where B occurs.

When should I use the binomial mode?

When an experiment has a fixed number of independent trials, each with two outcomes and the same success probability, and you want the probability of a certain number of successes.

Can I enter percentages or fractions?

Yes. Probabilities may be typed as decimals (0.25), fractions (1/4) or percentages (25%). Outcome counts in single-event mode must be whole numbers.

What is the difference between odds and probability?

Probability compares favorable outcomes with all outcomes; odds compare favorable outcomes with unfavorable ones. A probability of 1/6 corresponds to odds of 1 : 5.

How do I calculate 'at least' in a binomial problem?

Choose 'P(X ≥ k) at least k'. The calculator adds the probabilities from k up to n and shows which terms it summed.

Work out your probability

Pick a problem type above and enter your numbers to get an exact probability with every step.

Use the Probability Calculator Ask the AI Math Solver

About the Probability Calculator

What is probability?

How to use the Probability Calculator

Formulas

Classical probability

Independent events

Conditional probability

Binomial probability

Worked example: Two dice: P(sum 7) = 6/36

Final answer

How we read your problem

Step-by-step solution

Formulas used

Detailed explanation

Independent events and the addition rule

Conditional probability

Binomial probability

Common mistakes

Tips

Frequently asked questions

How do I calculate the probability of an event?

How do I find the probability of two independent events both happening?

What is conditional probability?

When should I use the binomial mode?

Can I enter percentages or fractions?

What is the difference between odds and probability?

How do I calculate 'at least' in a binomial problem?

Work out your probability

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