Free Probability Calculator

What Is a Probability Calculator?

Quick answer: A probability calculator divides the outcomes that count by all possible outcomes, which is how you calculate probability as a number from 0 to 1. This one handles a single event. Enter both numbers, press Calculate, and read the chance that it happens and the chance that it doesn't. Try it with your own numbers.

It works as a chance calculator for one event at a time. It suits setups where every outcome is equally likely, such as a fair coin, a fair die, a standard deck of 52 cards or marbles in a bag, and results you counted yourself, like 28 heads in 50 flips. The tool just divides one number by another, so it can't tell a fair die from a loaded one. Check that assumption before you type.

It doesn't show percentages, fractions or odds, and it has no separate boxes for two events. The sections below cover a workaround for each.

How to Use This Probability Calculator

Two boxes and one button. The labels are worded awkwardly, so here is the map from label to meaning, spelling included.

1. Count the total. Type the number of possible outcomes into the field labeled "No of possible outcome". A coin is 2, rolling a die is 6 and a deck of cards is 52.
2. Count what qualifies. Type the number of outcomes that count as your event into "No of possible event occured", for example 4 when the event is drawing an ace.
3. Press Calculate. Pressing Enter inside either field does the same.
4. Read the Result card. It has two rows, and the row names promise counts but deliver probabilities.

The first row, "No of possible event that occured", is the probability that your event happens. The second, "No of possible event that do not occured", is the probability that it doesn't. Despite the wording, both rows are numbers from 0 to 1, not counts: type 6 and 1 and you get 0.17 and 0.83.

Use whole numbers above zero, with the total larger than the event count. The card won't refresh while you type, so press Calculate again after every change. An error message means nothing new was calculated, whatever the card still shows.

The copy icon beside each row copies the rounded value shown. The math runs in your browser, and in our tests the numbers weren't sent to our server.

Do this now: type 52 in the first box and 4 in the second, press Calculate and look for 0.08 and 0.92. Then type your own two numbers into the calculator above.

Probability Calculator Examples: Coin, Dice and Cards

Every row below was typed into the calculator and then checked against the exact fraction.

You do the counting and nothing checks it: enter 13 for face cards instead of 12 and you get a tidy, wrong 0.25. A standard deck has 13 ranks in 4 suits, so 52 outcomes, and the jack, queen and king of each suit make 12 face cards. Count outcomes, not categories: a face card is one category but 12 outcomes.

Two dice have 36 outcomes, not 21, because a 1 on the first die with a 6 on the second is a different outcome from the reverse. Treat the dice as distinct and all 36 pairs are equally likely. That's why a total of 7 has 6 ways out of 36, which makes it the most common sum of two dice, while a total of 2 has exactly one.

As a dice probability calculator it needs only a total, 6 for one die or 36 for two, plus the count your question calls for. Rows two, three and eight are the dice examples. The fourth column is what the Result card prints, rounded, and the last is the exact fraction, so a small difference is only rounding, not an error. Change one count in the calculator and watch both numbers move.

How to Calculate Probability: The Formula Behind the Results

The probability formula fits on one line: P(event) = favorable outcomes / total outcomes. That ratio is the probability of an event, or its likelihood. "Favorable" only means counted in your event, not lucky.

An outcome is one result, like the king of spades, while an event is any set of results you care about, like drawing a king. The denominator is the size of the sample space, the full list of things that could happen. The formula holds only when those outcomes are equally likely.

Every result is a probability between 0 and 1. A value of 0 marks an event that cannot happen, and 1 marks one that always does. The complement of an event is everything else, and the complement rule says its chance is 1 minus the chance that the event happens. Before rounding, the two numbers on the Result card therefore add to 1.

Take the aces: four of the 52 cards qualify, so 4 divided by 52 gives 0.0769, or 1/13 once reduced. The calculator shows 0.08 for the ace and 0.92 for everything else, because 1 minus 0.0769 is 0.9231.

Inside the page, the script reads both boxes as numbers, divides the event count by the total, and subtracts that quotient from 1 for the second row. Each row is then rounded with JavaScript's Math.round(x * 100) / 100. Nothing else happens behind the button, so you can rebuild any answer on paper, give or take a hundredth on the exact ties covered below.

For the formal definitions, see the free textbook section OpenStax Introductory Statistics, terminology.

Why Small Probabilities Show as 0

The Result card rounds each value to two decimal places and drops trailing zeros, so you see 0.5 and 1 instead of 0.50 and 1.00. Anything rarer than about 1 in 200 runs out of room: the value falls under 0.005 and displays as 0.

In our tests 1 out of 1,000 showed 0 and 1, and so did 1 out of 365. Two aces in a row, 12 out of 2,652, also showed 0 and 1, although the exact value is 0.0045. Don't read a 0 as impossible. Divide the two counts yourself, or use the percentage route in the next section, which prints more digits.

Each value is rounded on its own, so a displayed pair can add to 1.01. Take 1 out of 8: the exact values 0.125 and 0.875 each sit halfway between two candidates, and both round up, to 0.13 and 0.88.

Ties can go either way. The value 23 out of 40 is exactly 0.575, yet the card shows 0.57 and 0.43, because browsers store decimals in binary and 0.575 * 100 returns 57.49999999999999 in a console. Its mirror, 17 out of 40, shows 0.43 and 0.57, since 0.425 * 100 lands on exactly 42.5. We scanned every total up to 2,000 and every event count below it. Of the 2,600 exact ties each row meets, 80 in the first row and 340 in the second rounded down.

Turn a Probability Into a Percentage, Fraction or Odds

This isn't a percentage chance calculator on its own, because the card prints decimals. Multiply by 100 to get a percent chance, so 0.17 reads as about 17 percent. The boxes don't take percentages either, only the two whole-number counts.

For more digits, open the Percentage Calculator, choose the formula "Y out of X is What%", type the outcomes that count first and the total second, and press Calculate. In our tests it answered to three decimals: 1 and 6 gave 16.667, 1 and 36 gave 2.778, and 12 and 2,652 gave 0.452. That one calculates on our server.

If the arithmetic feels rusty, our guide on how to calculate a percentage walks through it. A fraction is just the two counts, reduced: 4 out of 52 is 1/13, and 3 out of 6 is 1/2. The calculator doesn't output fractions, so reduce the two counts by hand.

Odds in favor compare the outcomes that count with the ones that don't, which makes a six on a die 1 to 5. To turn odds of a to b into a probability, use a divided by (a plus b), so 1 to 5 becomes 1/6. Going the other way, a probability p becomes odds of p to (1 minus p), which turns 0.25 into 1 to 3.

Odds are not counts: odds of 1 to 5 describe a total of 6 and an event count of 1, but typing 5 and 1 shows 0.2 and 0.8, the answer to a different question. A six on a die is 1/6, 0.17 on the card and odds of 1 to 5, so if your own conversion disagrees, recheck which count you divided by.

Theoretical vs Experimental Probability

Theoretical probability comes from the setup, before anything happens: assume the outcomes are equally likely and count. A fair die gives 1 out of 6, so you enter 6 and 1 and get 0.17. Experimental (empirical) probability comes from what did happen. It is the number of times the event occurred divided by the number of trials, which some textbooks call the relative frequency.

The same two boxes cover both. For an experiment, the total is the number of trials and the event count is how many times it happened. Flip a coin 50 times, get 28 heads, enter 50 and 28, and the card shows 0.56 and 0.44 against a theoretical 0.5. Trials go first and the number of times it happened goes second; swap them and you'll get the error message instead of a result.

A gap of 0.06 is ordinary at this size. The exact binomial sum puts 28 or more heads in 50 fair flips at about 24 percent, and a miss that big in either direction at nearly half of all runs. The typical wobble in the proportion, the standard error, is about 0.07 at 50 flips and 0.022 at 500, so ten times the flips narrows it by a factor of about three.

This pattern is the law of large numbers: more trials tend to pull the experimental value toward the theoretical one, but any single run can still stray.

A persistent gap is also how you catch a bent coin or a loaded die: the formula assumes fairness, and the experiment reports what happened. A fair coin lands 70 or more heads in 100 flips about once in 25,000 runs (enter 100 and 70 and the card shows 0.7 and 0.3), and the 1/2 formula can't warn you, because it assumes the coin is fair. For exercises, try these free probability practice problems.

Sample size: treat a handful of trials as a hint, not a measurement. 28 heads in 50 flips and 280 in 500 print the same 0.56, but a rough 95 percent margin of error is about 14 points for the first and about 4 for the second.

Probability of Two Events and Multiple Events

This calculator does one event at a time. For two or more events, combine the counts first, then enter the combined counts once.

Never multiply the rounded results. Two sixes on two dice would be 0.17 times 0.17, which is 0.0289, but the exact value is 1/36, or 0.0278, so the shortcut overshoots by about four percent.

Under the multiplication rule, two independent events that must both happen mean multiplying the totals together and the event counts together. The addition rule covers either of two events that can't overlap (mutually exclusive): keep the total and add the event counts. If they can overlap, add and then subtract the overlap once.

Forget the overlap and the ace of hearts gets counted twice: entering 17 instead of 16 in the heart-or-ace row shows 0.33 instead of 0.31.

Draws without replacement are dependent events: each draw shrinks the deck, so the second card comes from 51, not 52. Two aces in a row is 4 times 3 favorable ordered pairs out of 52 times 51, which is 12 out of 2,652. The card shows 0 and 1 for that, since the exact value is 1/221, about 0.45 percent, so take the percent from the Percentage Calculator route above. Put the first card back and reshuffle, which is drawing with replacement, and the count becomes 16 out of 2,704, or 1/169.

For at least one, work backwards from 1 minus the chance of none. A single roll misses six 5 times out of 6, so three rolls in a row miss it 125 times out of 216. Enter 216 and 125 and the card's second row, 0.42, is the chance of at least one six (exactly 91/216, about 0.4213). Counting those 91 sequences directly means adding 75 with one six, 15 with two and 1 with three, so the complement row is the cheaper route.

Conditional probability, Bayes' rule and the binomial and normal distributions need other calculators. The textbook's section on addition and multiplication rules covers the formal versions.

What This Probability Calculator Cannot Do

Five limits worth knowing before you rely on a result.

• One event only. There are no boxes for event A and event B, and no conditional probability, Bayes' rule, or binomial, normal and Poisson distributions. It shows no working either, so check your counts on paper first.
• Two decimals, nothing else. You get no percent, fraction or odds, and the copy icon copies the rounded value, so it can't rescue a 0.
• Whole numbers above zero. Zero or a negative number in either box shows "Invalid Input". An event count equal to or larger than the total shows "Outcome must be greater than the events occured". That rules out both an impossible event (0 outcomes, probability 0) and a certain event (all outcomes, probability 1). Your browser also flags a decimal in either field, because counts are meant to be whole.
• Equally likely outcomes. A loaded die, a stacked deck or a streaky shooter breaks the formula, not the calculator.
• Long-run frequency only. A probability describes a fair setup over many repetitions. It doesn't predict the next draw, and it isn't advice for games played for money.

Who Uses a Probability Calculator?

An online probability calculator fits problems you can count, where every outcome is equally likely.

• Math homework. Students test their answers on coins, dice and cards, then compare the display with the exact fraction.
• Classroom examples. Teachers can pair a counted experiment with the theoretical value in a statistics lesson.
• Game nights. Players settle arguments in board games and card games with fair dice and a shuffled deck, where the equally likely assumption holds.
• Counted results. A defect rate is an experimental probability: 3 faulty items in a sample of 20 is 20 and 3, which shows 0.15 and 0.85.

Related Free Calculators

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• all free online calculators: the full list in one place.