Probability Calculator

How to Use the Probability Calculator

This probability calculator helps you find probabilities involving two events, repeated independent events, or solve for an unknown probability from values you already know using a general probability solver option.

Two Events

If your probability problem involves two events A and B:

1. Choose the Two Events option
2. Select the relationship between A and B. Here, you need to specify whether the events are independent, mutually exclusive, or General events.
3. Select the probability you want to find.
4. Enter P(A) and P(B) as decimals. If you selected General events, you will also be required to enter P(A ∩ B).
5. Click Calculate.

The calculator will instantly return the correct probability value, a Venn diagram showing the representation of the answer, and a clear, step-by-step solution of how the value was obtained. The two events option will help you compute the following probabilities:

• Probability that both A and B occurred, P(A ∩ B)
• Probability of A or B, P(A ∪ B)
• Probability of A only
• Probability of B only
• Probability of exactly one of A or B
• Probability of Neither A nor B
• Probability of not A happening
• Probability of not B happening

Repeated Events

The repeated events option of our probability calculator allows you to calculate the probability that event A occurs every time, A occurs at least once, or A never occurs. This comes in handy when the same independent event can occur several times, and its probability stays the same on every trial.

To use this option of the calculator:

1. Enter the probability of A on one trial, P(A).
2. Enter the number of independent trials, n.
3. Select whether you want to find the probability that A occurs every time, never, or at least once.
4. Click Calculate.

The calculator will return the correct probability and show you exactly how the value was calculated, step-by-step.

Note. If your question asks for exactly, at most, or at least a specific number of successes, use our Binomial Probability Calculator instead.

Probability Solver

The probability solver option of the calculator helps you find P(A), P(B), P(not A), P(not B), P(A ∩ B), and P(A ∪ B) when related probabilities are known. This option is ideal when you need to find a missing probability, when you are given information about other related probabilities.

To use this option of the calculator:

1. Select the Probability solver option
2. Select the probability you want to find
3. Choose the calculation method that matches the already known information in your question
4. Enter the known probabilities as decimals
5. Click calculate

The calculator will instantly return the correct probability, the corresponding Venn diagram, and a clear, step-by-step solution of how to find the probability manually.

What Is a Probability Calculator?

A probability calculator helps you calculate how likely an event or combination of events is to occur. Probability values range from 0 to 1:

• A probability of 0 means the event cannot occur.
• A probability of 1 means the event is certain.
• Values between 0 and 1 describe different levels of likelihood.

For a simple experiment with equally likely outcomes, probability can be defined as: P(A) = number of favorable outcomes (X) / total number of possible outcomes (N)

However, many probability problems involve more than one event. In those cases, the correct calculation depends on whether the events are independent, mutually exclusive, or allowed to overlap.

Our probability calculator is designed primarily for those two-event and repeated-event problems.

Probability of Two Events

Suppose A and B are two events. We may want to know whether both occur, whether at least one occurs, whether exactly one occurs, or whether neither occurs.

Common probability notation includes:

Before calculating a two-event probability, the most important step is to determine how A and B are related.

Independent, Mutually Exclusive, and General Events

The relationship between the two events determines how we find their overlap, P(A ∩ B).

Independent Events

Two events are independent when the occurrence of one event does not change the probability of the other. For example, suppose you roll a fair die twice. Getting a 6 on the first roll does not change the probability of getting a 6 on the second roll. Thus, the two events are independent.

For independent events, P(A ∩ B) = P(A) × P(B)

Mutually Exclusive Events

Mutually exclusive events cannot occur at the same time. In other words, if two events are mutually exclusive, the occurrence of one completely rules out and prevents the occurrence of the other.

For example, on one roll of a die:

• A = rolling a 2
• B = rolling a 5

A single roll cannot be both 2 and 5, so the events are mutually exclusive.

For mutually exclusive events A and B, P(A ∩ B) = 0 because A and B cannot happen at the same time.

General Events

General events may overlap, but we should not assume that they are independent. In this case, P(A ∩ B) must be known or calculated from other information.

For instance, a college student may belong to a study group and also use the tutoring center. The events can occur together, so their overlap may be greater than 0.

If your problem gives a conditional probability such as P(A|B) instead of the intersection, use the Conditional Probability Calculator instead.

How to Find the Probability of A and B

The probability that both A and B occur is called the intersection and is written: P(A ∩ B)

How you calculate it depends on the relationship between the events. If the two events are independent: P(A ∩ B) = P(A) x P(B). However, if the events are mutually exclusive (dependent), P(A ∩ B) = 0

Therefore, for general events, the joint probability must be known or obtained from additional information.

Example 1: Probability of Two Independent Events

A university survey found that the probability a randomly selected student uses the campus recreation center in a given week is 0.40. The probability that the same student attends a university-sponsored lecture is 0.30. Assume the events are independent. Find the probability that the student does both.

Solution

Let:

• A = student uses the recreation center
• B = student attends the lecture

From the question, we know that:

• P(A) = 0.40
• P(B) = 0.30
• A and B are independent

We want to find P(A ∩ B).

Since the events are independent:

P(A ∩ B) = P(A) x P(B)

Substituting the values into the formula, we get:

P(A ∩ B) = 0.40 × 0.30

= 0.12

Therefore, P(A ∩ B) = 0.12. This implies that the probability that the student both uses the recreation center and attends the lecture is 0.12.

How to Find the Probability of A or B

The probability that A or B or both occur is called the union and is written: P(A ∪ B)

For events that may overlap, use the addition rule of probabilities. In this case, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). As you can see, we subtract P(A ∩ B) because the overlap is included once in P(A) and once again in P(B).

However, if A and B are mutually exclusive, their intersection is 0. The formula then becomes: P(A ∪ B) = P(A) + P(B)

Example 2: Mutually Exclusive Events

A college randomly selects one student from a graduating class. The probability that the student receives the economics department award is 0.08, while the probability of receiving the mathematics department award is 0.06. Assume the rules allow a student to receive only one of these two awards. Find the probability that the selected student receives either award.

Solution

Let:

• A = student receives the economics award
• B = student receives the mathematics award

From the question:

• P(A) = 0.08
• P(B) = 0.06
• A and B are mutually exclusive

Since the events cannot occur together, P(A ∩ B) = 0. Therefore, we can calculate the union as follows:

P(A ∪ B) = P(A) + P(B)

= 0.08 + 0.06

= 0.14

Therefore, P(A ∪ B) = 0.14. This implies that the probability that the selected student receives either award is 0.14.

How to Calculate General Overlapping Events

When A and B may overlap, and independence cannot be assumed, you need P(A ∩ B).

Once the intersection is known, the calculator can find all the related probabilities.

Suppose:

• P(A) = probability of A
• P(B) = probability of B
• P(A ∩ B) = probability of both

Then:

• P(A only) = P(A) − P(A ∩ B)
• P(B only) = P(B) − P(A ∩ B)
• P(exactly one) = P(A only) + P(B only)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• P(neither) = 1 − P(A ∪ B)

Example 3: General Overlapping Events

A university survey found that 46% of students participate in a study group and 38% use the tutoring center. In addition, 20% do both. Find the probability that a randomly selected student does exactly one of these activities.

Solution

Let:

• A = student participates in a study group
• B = student uses the tutoring center

From the question, we know that:

• P(A) = 0.46
• P(B) = 0.38
• P(A ∩ B) = 0.20

Since the events may overlap and the joint probability is given, these are general events. To find P (exactly one), follow these steps:

Step 1. Find P(A only)

Recall. P(A only) = P(A) − P(A ∩ B)

Substituting the values, we get:

Find P(A only) = 0.46 − 0.20

= 0.26

Step 2. Find P(B only)

Recall. P(B only) = P(B) − P(A ∩ B)

Substituting the values and solving gives:

P(B only) = 0.38 − 0.20

= 0.18

Since we have the required probabilities, we can calculate P(exactly one). Exactly one event occurs when either A only or B only occurs.

Therefore, P(exactly one) = P(A only) + P(B only)

= 0.26 + 0.18

= 0.44

Therefore, P(exactly one) = 0.44. This means that the probability that the student participates in exactly one of the two activities is 0.44.

Understanding the Venn Diagrams

The Venn diagrams in the calculator show which part of the sample space belongs to each probability. For two events, the circles represent A and B, while the surrounding rectangle represents the full sample space. The shaded region changes depending on the probability selected.

The table below provides a quick summary of how to shade various probabilities in a Venn diagram.

These diagrams are particularly useful when deciding whether a probability question is asking for AND, OR, exactly one, or neither.

A and B vs. A or B

The words and and or describe different regions of a probability problem. “A and B” means both events occur and is represented by P(A ∩ B). In a Venn diagram, it is the overlapping part of A and B.

“A or B” means A occurs, B occurs, or both occur. It is represented by P(A ∪ B). In probability, “or” normally includes the possibility that both events occur unless the problem specifically says “but not both.”

What Does Exactly One of A or B Mean?

Exactly one means that either A occurs or B occurs, but not both. You can find it by adding the A-only and B-only regions. In other words, P(exactly one) = P(A only) + P(B only). An equivalent formula is P(exactly one) = P(A) + P(B) − 2P(A ∩ B)

Suppose:

• P(A) = 0.50
• P(B) = 0.40
• P(A ∩ B) = 0.15

Then:

P(exactly one) = 0.50 + 0.40 − 2(0.15)

= 0.90 − 0.30

= 0.60

Therefore, P(exactly one) = 0.60

How to Find the Probability of Neither A nor B

“Neither A nor B” means that A does not occur and B does not occur. To find the probability that neither A nor B occurs, follow these steps:

1. Find the probability that A or B occurs using the formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
2. Use the complement rule: P(neither) = 1 − P(A ∪ B)

For example, if P(A ∪ B) = 0.72, then P(neither) = 1 − 0.72 = 0.28

Complement Probability

The complement of an event is the probability that the event does not occur.

For event A, P(not A) = 1 − P(A). Similarly, P(not B) = 1 − P(B)

For instance, if P(A) = 0.64, P(not A) = 1 − 0.64

= 0.36

Tip. An event and its complement always add up to 1.

Probability of Repeated Independent Events

Repeated-event probability applies when the same type of event is observed several times under independent conditions. For repeated independent events, the following two requirements must be satisfied:

• each trial must be independent;
• the probability of A must remain the same on every trial.

The calculator can find the probability that A occurs every time, never occurs, or occurs at least once.

If A has probability P(A) and there are n independent trials:

• P(A every time) = [P(A)]ⁿ
• P(A never occurs) = [1 − P(A)]ⁿ
• “At least once” is easiest to calculate using the complement of never occurring. Thus, P(at least one A) = 1 − [1 − P(A)]ⁿ

Example 4: Probability an Event Occurs Every Time

A university’s online system successfully processes a particular type of submission with probability 0.80 on any independent attempt. Assume three submissions are processed independently. Find the probability that all three submissions are processed successfully.

Solution

From the question:

• P(A) = 0.80
• n = 3

To find the probability that A occurs on every trial, we use the formula: P(A every time) = [P(A)]³

= 0.80³

= 0.512

Therefore, P(A every time) = 0.512. This means that the probability that all three submissions are successfully processed is 0.512.

Example 5: Probability an Event Never Occurs

A student has a 0.30 probability of receiving a particular optional survey question on each of four independent practice quizzes. Find the probability that the student never receives the question.

Solution

From the question:

• P(A) = 0.30
• n = 4

To find the probability that the student never receives the question, we use the formula: P(A never occurs) = [1 − P(A)]ⁿ

Substituting the values, we get:

P(A never occurs) = [1 − 0.30]⁴

= 0.70⁴

= 0.2401

Therefore, P(A never occurs) = 0.2401. This implies that the probability that the student never receives the question in the four quizzes is 0.2401.

Example 6: Probability of At Least One Event

Suppose a student has a 0.15 probability of being selected for a classroom presentation in each of five independent class meetings. Find the probability that the student is selected at least once.

Solution

From the question:

• P(A) = 0.15
• n = 5

To find the probability that the student is selected at least once, use the formula: P(at least one A) = 1 − [1 − P(A)]ⁿ

Substituting the values and solving gives:

P(at least one A) = 1 − [1 − 0.15]⁵

= 1- 0.85⁵

= 1 − 0.443705

P(at least one A) = 0.556295

Therefore, the probability that the student is selected at least once during the five meetings is approximately 0.556295.

Tip. If you need the probability of exactly 2 successes, at most 3 successes, or another specific number of successes, use our Binomial Distribution Calculator.

How to Use the Probability Solver

The probability solver works backward from probabilities you already know. Instead of selecting the relationship between A and B first, choose the probability you want to find. The calculator then asks for the information needed to solve for that value.

It can solve for:

• P(A)
• P(B)
• P(not A)
• P(not B)
• P(A ∩ B)
• P(A ∪ B)

This is useful when a homework or exam problem gives the union but asks for the intersection, or gives a complement and asks for the original probability.

Example 7: Find P(A ∪ B)

Suppose:

• P(A) = 0.52
• P(B) = 0.36
• P(A ∩ B) = 0.18

Find P(A ∪ B).

Solution

Use the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

= 0.52 + 0.36 − 0.18

= 0.70

Therefore, P(A ∪ B) = 0.70

Example 8: Find P(A ∩ B) When the Union Is Known

At a university, suppose:

• P(A) = 0.48
• P(B) = 0.41
• P(A ∪ B) = 0.72

Find P(A ∩ B).

Solution

Recall. P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Rearrange the formula, P(A ∩ B) = P(A) + P(B) − P(A ∪ B)

Substituting the values and solving, we get:

P(A ∩ B) = 0.48 + 0.41 − 0.72

= 0.17

Therefore, P(A ∩ B) = 0.17

Example 9: Find P(A) From Other Known Probabilities

Suppose:

• P(A ∪ B) = 0.68
• P(B) = 0.35
• P(A ∩ B) = 0.14

Find P(A).

Solution

Recall. P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Solving for P(A) gives: P(A) = P(A ∪ B) − P(B) + P(A ∩ B)

Substituting the values and solving, we have:

P(A) = 0.68 − 0.35 + 0.14

= 0.47

Therefore, P(A) = 0.47

When Should I Use Bayes’ Theorem?

Bayes’ theorem is useful when you want to reverse a conditional probability using prior information and new evidence. For example, you may know P(B|A) but want to find P(A|B). In that case, use our Bayes Theorem Calculator.

Probability vs. Probability Distribution

The calculator on this page works with probability rules involving events. A probability distribution is different. It describes how probabilities are assigned across possible values of a random variable.

For example:

• the binomial distribution models the number of successes in a fixed number of independent trials;
• the Poisson distribution models counts of events over an interval;
• the normal distribution models many continuous measurements.

If your problem specifies a probability distribution, use the appropriate calculator from our collection of Probability Distribution Calculators.

Using Combinations and Permutations in Probability

Some probability questions require you to count the number of possible outcomes before calculating the probability. Use a Combination Calculator when order does not matter or a Permutation Calculator when order matters.

Once you know the number of favorable and possible outcomes, you can use those counts to set up the required probability.

Common Probability Mistakes

Most errors in two-event probability problems come from choosing the wrong rule rather than performing the arithmetic incorrectly.

Watch for these common mistakes:

• Assuming events are independent without being told. If independence is not justified, do not automatically multiply P(A) and P(B).
• Confusing independent with mutually exclusive. Independent events can occur together, while mutually exclusive events cannot.
• Adding P(A) and P(B) without subtracting the overlap. For overlapping events, use P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Treating “or” as exactly one. In probability, A or B normally includes the case where both occur.
• Using the grand total for a conditional probability. Once an event is given, the relevant sample space changes.
• Using the repeated-events formulas when trials are dependent. These formulas require independent trials with the same probability on each trial.
• Entering percentages in this calculator. Enter probabilities as decimals. For example, enter 35% as 0.35.

Frequently Asked Questions