Probability Calculator

Probability calculator at a glance#

How do you calculate the probability of an event?#

To find the probability of an event when every outcome is equally likely, you divide the number of favorable outcomes by the total number of outcomes. For instance, rolling a 6 with one die is 1 out of 6, so the probability is 1/6, which is about 0.1667 or 16.7%. If you're drawing a heart from a 52-card deck, that's 13 out of 52, equaling 0.25. The calculator above requires probabilities like these in decimal form between 0 and 1 to determine the result of combining two events.

What is the probability of two events both happening?#

When dealing with independent events, you multiply the probabilities. For example, the chance of rolling an even number (0.5) and drawing a heart (0.25) is 0.5 × 0.25 = 0.125, or 1 in 8. Enter 0.5 and 0.25 in the Probability of Two Events mode. The calculator provides this result along with the probabilities of either, exactly one, neither, and each event occurring without the other.

What is experimental probability?#

Experimental probability, also known as empirical probability, is calculated by dividing the number of times an event occurred by the total number of trials. If a T-shirt stand sold 6 purple shirts and 9 shirts in other colors, then the experimental probability of selling a purple shirt next is 6 out of 15, or 0.4. After dividing the counts, enter the decimal in the calculator.

Probability basics#

Probability ranges from 0, indicating an event that cannot occur, to 1, indicating an event that is certain. According to OpenStax Introductory Statistics, you calculate the probability of an event with equally likely outcomes by counting the event's outcomes and dividing by the total number of outcomes.

The same probability can be expressed as a fraction, decimal, or percent: 1/4, 0.25, and 25% all describe drawing a heart. The calculator requires the decimal form. To convert 1/6 into a decimal, divide 1 by 6 and enter 0.1666667, or use the fraction to decimal calculator. For a percentage chance, divide by 100, so a 15% chance becomes 0.15.

The complement of an event covers everything outside it. The chance that event A does not occur is P(A') = 1 − P(A), meaning the probability of not drawing a heart is 1 − 0.25 = 0.75.

Two events: and, or, not#

Consider two separate actions: rolling a die and drawing a card from a full deck. Event A is rolling an even number on the die, with a probability of 3/6 = 0.5. Event B is drawing a heart, with a probability of 13/52 = 0.25. When combined, the die and the card create 6 × 4 = 24 equally likely combinations of numbers and suits, displayed in the grid below.

Entered as 0.5 and 0.25, the Probability of Two Events mode provides eight results:

• Not A: 1 − 0.5 = 0.5. Not B: 1 − 0.25 = 0.75.
• A and B: 0.5 × 0.25 = 0.125, representing the 3 shaded cells where an even number coincides with a heart.
• A or B, or both: 0.5 + 0.25 − 0.125 = 0.625, referring to the 15 cells that are even, a heart, or both.
• Exactly one of them: 0.5 + 0.25 − 2 × 0.125 = 0.5.
• Neither: 1 − 0.625 = 0.375.
• A but not B: 0.5 × 0.75 = 0.375. B but not A: 0.5 × 0.25 = 0.125.

The formula for A or B subtracts the overlap because adding 0.5 and 0.25 counts the 3 even hearts twice. OpenStax labels this the addition rule, and the A and B result is its multiplication rule for independent events.

When the two events are not independent#

The calculator assumes A and B are independent, meaning one event doesn't affect the chance of the other. A die and a card are independent, but two events on the same die are not. An even number and a 4 or higher each have a probability of 0.5 in one roll, but both occur only on a 4 or a 6, which is 2 out of 6, approximately 0.333 instead of the 0.25 the calculator would predict.

Drawing cards without replacement introduces dependency. The chance of two aces in a row is 4/52 × 3/51 = 12/2652, which simplifies to 1/221, about 0.0045. The second fraction has 3 aces in 51 cards because the first ace is no longer in the deck. The calculator doesn't accommodate this type of conditional probability, so manual calculation is necessary.

Working backward from two known probabilities#

The Probability Solver for Two Events mode can work with any two of eight probabilities: A, B, not A, not B, A and B, A or B, exactly one, and neither. It calculates the remaining values, assuming the events are independent. By entering 0.125 as the probability of both events and 0.625 as the probability of either, the solver can determine the pair 0.25 and 0.5. It doesn't identify which event is A, so it lists the smaller value as P(A).

Certain pairs don't provide enough information. A probability and its complement, like P(A) and P(A'), count as a single fact. The calculator also can't resolve P(A), P(B), or their complements when paired with the probability that exactly one event occurs, resulting in an error for those inputs.

Repeated tries#

The Series of Independent Events mode requires a probability and a number of repeats for two events, A and B (fill in both). For each event, it calculates the probability of it occurring every time, never occurring, and occurring at least once. It then combines A and B in the same manner.

Consider an event with a 1% probability on each attempt, repeated 25 times. Enter 0.01 and 25 for A, with any values in the B boxes. The probability it never occurs is 0.99²⁵ = 0.7778214, making the probability it occurs at least once 1 − 0.7778214 = 0.2221786, approximately 22%. Small risks can accumulate. The chart below illustrates how the probability of at least one occurrence increases with more attempts.

A coin toss follows the same logic. With a probability of 0.5 and 4 repeats, the probability of getting 4 heads in a row is 0.0625, while the probability of at least one head is 0.9375.

Normal distribution mode#

The fourth mode is designed for measurements that align with a bell curve. You enter the mean and standard deviation, along with a left and right bound. For an open end, you can type -Infinity or Infinity. The calculator will provide the probability within the bounds, outside them, at or below the left bound, and at or above the right bound. It also details the range around the mean for 11 confidence levels, ranging from 0.6828 to 0.99999.

Consider test scores with a mean of 70 and a standard deviation of 8. The probability of a score falling between 62 and 78 is 0.68268. Similar to the z-score calculator, each bound is converted into a z-score, rounded to two decimal places, and then the area is read from a standard normal table.

Where these rules come up#

• Games of chance such as dice totals, card draws, and coin flips are classic examples of equally likely cases, and you can see common ones in the table below.
• Repeated risks highlight how a small chance per try can accumulate over many tries, demonstrated by the at least once rule.
• Odds: chances in lotteries and games are often quoted as odds rather than probabilities. For a probability of 0.25, the odds are 1 to 3 in favor, and the odds calculator turns odds into a probability.
• Counting outcomes becomes necessary when listing favorable and total outcomes is impractical. The permutation calculator aids in counting ordered arrangements.

Common dice and card probabilities#

Each row assumes a fair six-sided die or a standard 52-card deck, with each outcome being equally likely. The two rows that involve events without replacement are dependent, so they require manual calculations.

Sources#

• OpenStax, Introductory Statistics 2e, section 3.1: Terminology
• OpenStax, Introductory Statistics 2e, section 3.3: Two Basic Rules of Probability

Frequently asked questions#

Can I enter a fraction like 1/6?#

No, each probability box requires a decimal between 0 and 1. You should divide first, so 1/6 becomes 0.1666667. The results display up to 7 significant digits, so rounding your input might change the last digit of an answer.

How do I turn a percent chance into a probability?#

You divide by 100. For instance, a 15% chance is 0.15 and a 0.2% chance is 0.002. A figure like 1 in 500 is 1 ÷ 500 = 0.002 as well.

What is the difference between A or B and exactly one?#

A or B includes the scenario where both occur, while exactly one excludes that. For the situation involving an even number and a heart, A or B is 0.625 and exactly one is 0.5. The difference is the 0.125 chance of both happening.

How do you find the probability of at least one?#

Calculate the chance that the event never occurs and subtract that from 1. The probability of at least one 6 in two rolls of a die is 1 − (5/6 × 5/6) = 11/36, which is about 0.3056.

What if the two events cannot happen together?#

Events that cannot occur simultaneously are mutually exclusive and not independent. The probability of both is 0, so you should add instead: rolling a 1 or a 2 on one die is 1/6 + 1/6 = 1/3. Avoid using the calculator's two-event modes for these situations, since they assume independence.

Can it find the chance of exactly 2 successes in 5 tries?#

Not directly. The series mode accounts for every time, never, and at least once. For exactly k successes in n attempts, apply the binomial formula C(n, k) × p^k × (1 − p)^{n − k}. For example, getting exactly two 6s in five rolls is 10 × (1/6)² × (5/6)³, about 0.1608.