Probability Calculator

Translate an event question into the right rule without assuming independence or double-counting overlap.

Drag & drop or upload an image or PDF

Enter a probability problem, such as draw 2 red balls without replacement from 5 red and 3 blue

Use fractions, decimals from 0 to 1, or percentages. Say whether events are independent, dependent, mutually exclusive, or conditional.

Try an example

Event relationships choose the rule

A probability is a number from 0 to 1 that measures how likely an event is under a stated model. Counts can be divided directly only when the outcomes in the sample space are equally likely.

For two events, words such as and, or, given, without replacement, and at least one carry mathematical meaning. The calculator names that relationship before it substitutes numbers, which prevents the most common addition and multiplication errors.

How to use the probability calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

The multiplication rule

The probability that both events occur equals the chance of the first event times the conditional chance of the second. For independent events, the conditional probability equals P(B).

P (A \cap B) = P (A) P (B ∣ A)

A reliable way to work through it

Define the event and sample space

List what counts as success and confirm whether the elementary outcomes are equally likely.

Name the relationship

Decide whether the problem involves a complement, union, intersection, conditional event, or independent repetition.

Use a check between 0 and 1

Simplify exact fractions and confirm the final probability is neither negative nor greater than 1.

Worked example

A bag has 5 red and 3 blue balls. Find the probability of drawing 2 red balls without replacement

P (R_{1}) = \frac{5}{8}

Five of the eight balls are red on the first draw.

P (R_{2} ∣ R_{1}) = \frac{4}{7}

After one red ball is removed, four red balls remain among seven total.

P (R_{1} \cap R_{2}) = \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14}

Use the conditional multiplication rule because the draws are dependent.

The probability of two red balls is 5/14, about 0.3571 or 35.71%.

Common mistakes to check

Assuming independence

Without-replacement draws and many conditional events change the probability of what happens next.

Adding overlapping events directly

For P(A or B), subtract P(A and B) unless the events are mutually exclusive.

Counting unequal outcomes as equal

Favorable outcomes divided by total outcomes works only when the underlying outcomes have equal probability.

Permutation and Combination CalculatorCount ordered arrangements and unordered selections, with the order decision made before the factorials.Z-Score CalculatorStandardize a value as a signed number of standard deviations from its mean.Standard Deviation CalculatorMeasure how far values spread from the mean, with the sample or population denominator shown clearly.Mean Median Mode CalculatorCompare three measures of center and see how sorting, frequencies, and outliers affect them.

Questions students ask

What is the probability of an impossible event?

It is 0. A certain event has probability 1, and every valid probability lies between those values.

When can I multiply two probabilities?

Use P(A)P(B) for independent events. For dependent events, use P(A)P(B given A) or the equivalent reversed form.

Are independent events the same as mutually exclusive events?

No. Independent events do not change each other's probabilities. Mutually exclusive events cannot occur together.

How do I find the probability of at least one event?

Often the quickest method is the complement rule: subtract the probability of none from 1.