Use this free probability calculator to find the probability of single events, two events, and probability distributions. Enter your values, and the tool instantly returns results for independent events, conditional probabilities, mutually exclusive events, and more.
Probability is the measure of the likelihood that an event will occur. It is always a number between 0 and 1, where 0 means the event cannot occur and 1 means it is certain. This calculator helps you determine the probability for common scenarios without manual math.
Use the Probability Calculator to Find the Probability of Event
To use the probability calculator, start by choosing what you want to solve. The tool covers single events, two events, and distribution calculations.
For a single event:
- Enter the number of favorable outcomes.
- Enter the total number of possible outcomes.
- The calculator returns the probability as a decimal, fraction, and percentage.
For two events:
- Enter the probability of Event A.
- Enter the probability of Event B.
- Select whether the events are independent, conditional, or mutually exclusive.
- The tool calculates the probability of A and B, the probability that either event occurs, and more.
You can also enter values for normal distribution, binomial distribution, or cumulative distribution calculations. Each mode asks for the specific inputs it needs, like mean, standard deviation, or number of successes.
Results are estimates for planning and learning. For professional statistical analysis, consult a qualified statistician.
Probability Definition and Probability Formula
The probability definition is straightforward. Probability is the ratio of the number of favorable outcomes to the total number of outcomes.
The basic probability formula:
P(A) = Number of favorable outcomes / Total number of possible outcomes
If you roll a fair six-sided die, the probability of rolling a 1 is 1/6, or about 16.67%. There is one favorable outcome (rolling a 1) and six possible outcomes.
A probability of 0 means the event cannot happen. A probability of 1 means it will definitely happen. Most real events fall somewhere in between, expressed as a decimal from 0 to 1, a fraction, or a percentage.
Theoretical vs. experimental probability
Theoretical probability is calculated based on the number of possible outcomes in a perfect model. Experimental (empirical) probability comes from actual trials. If you flip a coin 100 times and get heads 53 times, the experimental probability of heads is 0.53. The theoretical probability remains 0.50. As the number of trials increases, experimental results tend to approach theoretical values.
Calculate Probabilities for Two Events
When you have two events, the calculation depends on the relationship between them. The key question is whether the events are independent, conditional, or mutually exclusive.
For two events A and B, the main results are:
- P(A and B): The probability of both A and B occurring together (joint probability).
- P(A or B): The probability that either A or B occurs (or both).
The formulas change based on how the events relate to each other.
Probabilities of Two Events When Events Are Independent
Two events are independent if the occurrence of one event does not affect the probability of the other. Flipping a coin and rolling a die at the same time are independent events.
When events are independent:
- P(A and B) = P(A) × P(B)
- P(A or B) = P(A) + P(B) − P(A and B)
Example: What's the probability of heads on a coin flip AND rolling a 6 on a die?
P(Heads) = 0.5 P(Six) = 1/6 ≈ 0.1667 P(Heads and Six) = 0.5 × 0.1667 = 0.0833
That's about an 8.33% chance.
Repeating a trial does not change the probability of each individual event when the events are independent. Each coin flip stays at 50/50 regardless of previous results.
Conditional Probabilities for Two Events
Conditional probability measures the probability of one event occurring given that another event has already happened. It is written as P(A | B), read as "the probability of A given B."
The conditional probability formula:
P(A | B) = P(A ∩ B) / P(B)
This applies when events are dependent, meaning the outcome of one event does affect the likelihood of the other.
Example: A bag holds 5 red marbles and 3 blue marbles. You draw one marble from the bag without replacing it, then draw a second. The probability of drawing a red marble second depends on what you drew first.
If you drew red first: P(Red second | Red first) = 4/7 If you drew blue first: P(Red second | Blue first) = 5/7
The conditional probability calculator on this page handles these calculations. Enter P(A), P(B), and P(A ∩ B) to get the result.
Probability Distribution
A probability distribution describes all possible outcomes and their associated probabilities. It maps every result in a set of possible values to a likelihood. Distributions are essential for statistical analysis beyond single events.
Normal Distribution
A normal distribution is a continuous probability distribution shaped like a symmetric bell curve. Most values cluster around the mean, and the spread is determined by the standard deviation.
Key properties:
- The curve is symmetric around the mean.
- About 68% of values fall within one standard deviation of the mean.
- About 95% fall within two standard deviations.
- About 99.7% fall within three standard deviations.
A normal distribution calculator (sometimes called a standard normal calculator) lets you find the area under the curve for a given range. Enter the mean, standard deviation, and the value you want to evaluate. The tool returns the probability that a randomly selected value falls below, above, or between specific points.
Binomial Distribution
A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes (success or failure).
You need three inputs:
- n: The number of trials.
- p: The probability of success on each trial.
- k: The number of successes you want to find the probability for.
Example: Flip a coin 10 times. What is the probability of getting exactly 6 heads?
Here n = 10, p = 0.5, k = 6. The binomial formula calculates this as approximately 0.2051, or about 20.5%.
A binomial distribution applies when trials are independent and the probability stays constant across trials.
Cumulative Distribution
A cumulative distribution function (CDF) gives the probability that a random variable takes a value less than or equal to a specified point. Instead of asking "what is the probability of exactly this outcome," it answers "what is the probability of this outcome or anything less?"
The difference between a probability distribution and a cumulative distribution:
- A probability distribution shows the likelihood of each individual outcome.
- A cumulative distribution shows the running total of probabilities up to a given value.
For example, in a binomial distribution with n = 10 and p = 0.5, the cumulative probability of getting 6 or fewer heads is the sum of probabilities for 0, 1, 2, 3, 4, 5, and 6 heads. This is useful when you need the probability of at least a certain number of successes or at most a certain number.
Probability of Rolling Dice and Single Events
Rolling dice is one of the most common probability examples. A standard six-sided die has six equally likely outcomes.
- Probability of rolling any specific number (like a 1): 1/6 ≈ 16.67%
- Probability of rolling an even number (2, 4, or 6): 3/6 = 0.5
- Probability of rolling a number greater than 4: 2/6 ≈ 0.3333
What's the probability of rolling 2 sixes on two dice?
Since each die is independent: P = 1/6 × 1/6 = 1/36 ≈ 0.0278, or about 2.78%.
What about getting 1, 2, 3, 4, 5, 6 when rolling six dice?
You need each die to show a different number. The probability of the first die showing any number is 6/6. The second must differ (5/6), the third (4/6), and so on.
P = (6/6) × (5/6) × (4/6) × (3/6) × (2/6) × (1/6) = 720/46656 ≈ 0.0154, or about 1.54%.
For any probability of a single event, use the basic formula: favorable outcomes divided by total number of outcomes. Enter those two numbers into the calculator and get your result instantly.
Probability of A and B for Independent Events
The probability of A and B both occurring is called the joint probability. When events are independent, multiply their individual probabilities.
P(A and B) = P(A) × P(B)
This works only when the events are independent. If the occurrence of one event changes the likelihood of the second, you need the conditional probability formula instead.
Can you calculate the probability of A and B not occurring?
Yes. The probability that neither A nor B occurs when the events are independent:
P(not A and not B) = (1 − P(A)) × (1 − P(B))
You can also find the probability of at least one event occurring:
P(at least one) = 1 − P(not A and not B)
This is often more practical than calculating every possible combination separately.
Probability and Statistics: Standard Deviation, Mean, and Distribution Calculator
Probability and statistics are closely connected. Understanding mean, standard deviation, and distributions helps you interpret probability results in context.
- Mean is the average of all values in a data set.
- Standard deviation measures how spread out the values are from the mean.
- Variance is the square of the standard deviation.
These values feed directly into distribution calculators. A normal distribution calculator, for instance, requires the mean and standard deviation to find the probability of any value range.
This probability calculator covers the most common distribution types. For specialized tools like a Poisson distribution calculator, chi-square calculator, confidence interval calculator, or sample size calculator, check the related calculators on this site.
Probability Solver for Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur at the same time. Drawing a heart and drawing a spade from a single card draw are mutually exclusive.
When events are mutually exclusive:
- P(A and B) = 0 (they cannot happen together)
- P(A or B) = P(A) + P(B)
This is the "OR" addition rule in its simplest form. You can add probabilities directly because there is no overlap.
If A and B are mutually exclusive, then P(A ∩ B) = 0. This is a key distinction from independent events, where P(A ∩ B) = P(A) × P(B) and can be greater than zero.
When can you add probabilities for OR?
You can always use P(A or B) = P(A) + P(B) − P(A and B). When events are mutually exclusive, the last term is zero, so it simplifies to straight addition. When events are not mutually exclusive, you must subtract the overlap to avoid counting it twice.
The probability solver on this page identifies whether the events are mutually exclusive based on your inputs and applies the correct formula automatically.
Frequently Asked Questions
How Do I Use the Probability Calculator?
Select your calculation type: single event, two events, or distribution. Enter the required values. For a single event, input the number of favorable outcomes and the total number of possible outcomes. For two events, enter each event's probability and specify the relationship (independent, conditional, or mutually exclusive). Click calculate to see results as a decimal, fraction, and percentage.
How Do I Find the Probability of Two Events?
Use the formulas for two events based on their relationship. For independent events, multiply the individual probabilities for P(A and B). For mutually exclusive events, add the probabilities for P(A or B). For dependent events, use the conditional probability formula. Enter your values into the calculator and it applies the correct formula.
What Is a Normal Distribution Calculator?
A normal distribution calculator finds the area under the bell curve for a given range of values. You enter the mean, standard deviation, and the value or range you want to evaluate. It returns the probability that a randomly chosen data point falls within that range. This is useful for any continuous probability distribution that follows a bell-shaped pattern.
How Do I Calculate Conditional Probabilities?
Use the formula P(A | B) = P(A ∩ B) / P(B). Enter the joint probability of A and B and the probability of B. The result tells you the likelihood of A occurring, given that B has already occurred. This applies when events are dependent and the outcome of one event does affect the probability of the other.
What Is the Difference Between a Probability Distribution and Cumulative Distribution?
A probability distribution shows the likelihood of each individual outcome. A cumulative distribution shows the probability of a result being less than or equal to a given value. Think of it as individual probabilities versus a running total. The cumulative distribution always reaches 1.0 at the maximum possible value.
Can This Probability Solver Handle Independent Events and Conditional Probabilities?
Yes. The calculator supports independent events, conditional probabilities, mutually exclusive events, and joint probabilities. Select the appropriate mode, enter your values, and the solver applies the correct formula. It also handles normal, binomial, and cumulative distributions.
How Do I Calculate the Probability Using the Probability Formula?
The basic probability formula is P(A) = number of favorable outcomes divided by the total number of possible outcomes. Identify how many outcomes satisfy your event, count the total possible outcomes, and divide. For example, the probability of drawing an ace from a standard deck is 4/52 = 0.0769, or about 7.69%. Enter these numbers into the calculator for instant results.