Probability Calculator and Binomial Distribution Calculator
Our probability calculator works out the chance of a single event, two events together or a number of successes over many trials. It doubles as a binomial distribution calculator with exact, at most and at least results. Pick a mode and type your numbers to see the answer.
What Probability Means
Probability is a number between 0 and 1 that describes how likely something is. A value of 0 means the event cannot happen and 1 means it is certain. Most people find percentages easier, so 0.25 is the same as a 25% chance.
Every probability calculator starts from the same idea. You count the outcomes that give the result you want and compare them with all the outcomes that could happen. The harder part is combining events, repeating trials and avoiding the traps that fool our intuition.
How to Use the Probability Calculator
Choose one of three modes from the menu at the top of the tool. Each mode shows its own inputs and its own set of results. The hint under the inputs reminds you which format to use.
- Single event: enter the favorable outcomes, the total outcomes and how many times you try.
- Two independent events: enter P(A) and P(B) as decimals from 0 to 1.
- Binomial distribution: enter the number of trials n, the chance of success p and the number of successes k.
Every result shows a decimal and a percentage, and it updates as you type. Very small chances appear in scientific notation, so 1.5e-8 means 0.000000015. You can switch modes at any time without losing the numbers you typed.
How to Calculate Probability of a Single Event
The basic formula divides favorable outcomes by total outcomes, assuming each outcome is equally likely. Rolling a six on a fair die is 1 out of 6, which gives 0.1667 or 16.67%. Drawing an ace from a full deck is 4 out of 52, and the probability calculator shows about 7.69%.
The Complement Rule
The chance that an event does not happen is 1 minus the chance that it does. For the die, that is 1 − 0.1667, or 83.33%. The complement rule is often the fastest way to solve problems that use the words "at least".
Repeated Tries
If you roll the die four times, the chance of at least one six is 1 − (5/6)⁴. The probability calculator gives 0.5177, or 51.77%, which is just over an even chance. The chance of four sixes in a row is only 0.0772%.
Gamblers in 17th century France bet on this exact game. A player called the Chevalier de Méré also bet on getting a double six in 24 rolls of two dice, and he kept losing. The chance of that is only 49.14%, and his puzzle led Pascal and Fermat to lay the foundations of probability theory.
Probability Rules at a Glance
The probability calculator applies a handful of rules that cover most school and everyday problems. Each one has a short formula you can also use by hand. The table below sums them up for two events A and B.
| Question | Rule | Formula |
| Not A | Subtract from one | 1 − P(A) |
| A and B (independent) | Multiply | P(A) × P(B) |
| A or B (independent) | Add, then remove the overlap | P(A) + P(B) − P(A) × P(B) |
| A or B (cannot happen together) | Add | P(A) + P(B) |
| A given B | Divide by P(B) | P(A and B) ÷ P(B) |
| At least once in r tries | One minus never | 1 − (1 − p)^r |
Two Events: And, Or and Neither
When two events do not affect each other, they are called independent. The chance that both happen is P(A) × P(B). The chance that at least one happens is P(A) + P(B) − P(A) × P(B), because otherwise the overlap would be counted twice.
Say there is a 30% chance of rain and a 20% chance your bus is late, and the two are independent. The probability calculator gives 6% for both, 44% for at least one and 56% for neither. The chance of exactly one of them is 38%.
Mutually Exclusive Events
Some events cannot happen together, such as rolling a 2 and a 5 on the same die. For mutually exclusive events, P(A and B) is 0 and P(A or B) is simply P(A) + P(B). The two event mode assumes independence, so add the numbers yourself in that case.
Conditional Probability
Conditional probability is the chance of A once you know that B has happened, written P(A | B). It equals P(A and B) ÷ P(B). In a deck of 52 cards, the chance that a card is an ace given that it is a heart is 1/13, because one of the 13 hearts is an ace.
Binomial Distribution Calculator
A binomial setting has a fixed number of trials, two possible results for each trial and the same chance of success every time. The trials must also be independent. Coin flips, free throws and pass or fail quality checks all fit this pattern.
The formula is P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). C(n, k) counts the ways to arrange k successes among n trials. Our binomial distribution calculator adds up these terms for you, which matters because "at most" and "at least" questions need many of them.
Reading the Five Results
- P(X = k) is the chance of exactly k successes.
- P(X < k) and P(X ≤ k) are the chances of fewer than k and at most k.
- P(X > k) and P(X ≥ k) are the chances of more than k and at least k.
- The mean np is the expected number of successes.
- The standard deviation √(np(1 − p)) shows how much the count usually varies.
The binomial distribution calculator fills in all five at once, so you never have to add a long list of terms by hand. The exact result is useful for a single outcome. The cumulative ones answer most exam and business questions.
Worked Binomial Examples
Ten Coin Flips
Flip a fair coin 10 times and the binomial distribution calculator gives 0.1172, or about 11.72%, for exactly 3 heads. The chance of at most 3 heads is 17.19%. By symmetry, the chance of at least 7 heads is also 17.19%.
Even the most likely result, exactly 5 heads, happens only 24.61% of the time. Many people expect a fair coin to land close to half heads almost every time. The binomial distribution calculator shows how often small samples drift away from that.
Defects in a Batch
A machine makes parts with a 5% defect rate and you inspect 20 of them. The chance that none is faulty is 35.85%, and the chance of at most one is 73.58%. That leaves a 26.42% chance of finding two or more defects in a normal batch.
The mean is 1 defect and the standard deviation is 0.97. A quality team can use these figures to decide when a count is unusual enough to stop the line. Enter n = 20, p = 0.05 and k = 1 in the binomial distribution calculator to check.
Free Throws
A player who makes 70% of free throws takes 5 shots. The binomial distribution calculator puts exactly 4 makes at 36.02% and 4 or more at 52.82%. On average the player scores 3.5 of the 5 shots.
Guessing on a Test
A student guesses all 10 answers on a quiz with four choices per question. Set n = 10, p = 0.25 and k = 5 in the binomial distribution calculator, and the chance of 5 or more right is only 7.81%. The chance of getting every answer wrong is 5.63%, which is lower than most people guess.
When the Binomial Model Does Not Fit
The binomial distribution calculator assumes the chance of success stays the same on every trial. That breaks when you draw without putting items back from a small group, such as picking cards from one deck. In that case the hypergeometric distribution gives the right answer.
Results with more than two outcomes, such as a die roll counted by face, need the multinomial model instead. For very large n, the normal curve gives a fair estimate when np and n(1 − p) are both at least 10. For up to 10,000 trials, though, the binomial distribution calculator on this page gives the exact figure.
Expected Value
Expected value is the average result you would get if you repeated something many times. Multiply each outcome by its chance and add the products. Use the probability calculator to find each chance first, then do the multiplication.
Say a game costs 10 to play and pays 50 if you roll a six. The expected value is 50 × 1/6 − 10, which comes to about −1.67 per game. A negative number means you lose money on average, which is how every casino game is built.
Probability in Everyday Life
Weather forecasts, insurance prices and medical test results all rest on probability. A 70% chance of rain means that on days with the same conditions, it rained on about 7 out of 10. Sports analysts, traders and quality engineers use the same math every day.
A probability calculator helps you check claims instead of trusting a gut feeling. Rare events happen more often than people expect. Running the numbers before a decision often changes the decision.
Probability and Odds
Odds compare the chances for and against an event, while probability compares the chances for with the total. A probability of 1/6 equals odds of 1 : 5 in favor, or 5 : 1 against. The single event mode shows odds next to the fraction so you can switch between the two.
Common Probability Mistakes
- The gambler's fallacy: after five heads in a row, the next flip is still 50%.
- Adding chances for repeated tries, since two tries at 30% give 51%, not 60%.
- Assuming events are independent when one affects the other, such as drawing cards without putting them back.
- Ignoring how many people are involved, which is why the birthday problem surprises so many.
The Birthday Problem
In a room of just 23 people, the chance that two share a birthday is 50.73%. It feels too high because we picture our own birthday matching someone else's. In fact, 23 people create 253 possible pairs, and each pair is another chance for a match.
Why Use Our Probability Calculator
Our probability calculator puts single events, two events and binomial trials on one page. It is also a binomial distribution calculator with five cumulative results. It shows decimals, percentages, fractions and odds, and it handles up to 10,000 trials. Pick a mode, type your numbers and calculate now.
Frequently asked questions
Is this probability calculator free?
Yes, it is free with no sign up and no limit.
How to calculate probability of at least one success?
Subtract the chance of zero successes from 1, which the tool does for you.
Can I use it as a binomial distribution calculator for large n?
Yes, it accepts up to 10,000 trials.
What inputs does the binomial distribution calculator need?
The number of trials n, the chance of success p and the number of successes k.
Does the probability calculator show odds?
Yes, the single event mode shows odds for and against.
What does P(X ≤ k) mean?
It is the chance of k or fewer successes in n trials.
Can the probability calculator handle dependent events?
The two event mode assumes independence, so use the conditional formula for dependent events.
What is the chance of rolling a six at least once in four rolls?
It is about 51.77%.
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