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Dice, coin and game probability calculators

Tools for games of chance and the probability puzzles that defeat intuition. There are calculators for coin and dice probabilities, simulators for flipping coins, rolling dice and drawing random numbers, roulette odds, and step-by-step answers to paradoxes such as Monty Hall and the birthday problem.

Which calculator do I need?

You have or wantUse
The chance of k heads in n flips, fair or biased coinCoin flip probability calculator
The chance of a run of k heads in a rowCoin toss streak calculator
The chance of rolling a given total with several diceDice probability calculator
The average damage or total of a roll like 4d6 drop lowestDice average calculator
The payout and house edge of a roulette betRoulette payout calculator
How many people it takes for a shared birthday to be likelyBirthday paradox calculator
Whether to switch doors in the Monty Hall gameMonty Hall problem calculator

Coins

Flip virtual coins and work out the chances of heads counts and streaks.

Dice

Probabilities, averages and virtual rolls for any combination of dice.

Casino games and random draws

The house edge on roulette bets, and fair random numbers for draws and raffles.

Probability puzzles and paradoxes

Classic puzzles with the exact answer, the reasoning, and a simulation to check it.

Calculators, simulators and puzzles

This category has three kinds of page. The calculators give exact probabilities: the coin flip calculator uses the binomial distribution, the dice calculator counts every combination of faces, and the dice average calculator gives the expected value and SD of a roll. The simulators (coin flipper, dice roller, random number generator) produce actual random outcomes, which is useful for games and for seeing how results scatter around the expected values. The puzzle pages explain the famous paradoxes and let you simulate them.

Worked comparison: dice sums

With two six-sided dice there are 36 equally likely outcomes. A total of 7 can be made 6 ways (1+6, 2+5, 3+4 and the reverses), so P(7) = 6/36 = 16.7%. A total of 2 or 12 can be made only one way each, so each has probability 1/36 = 2.8%. The average total is 7 and the standard deviation is 2.42. With three dice the most likely totals are 10 and 11, each at 27/216 = 12.5%.

Puzzle answers at a glance

PuzzleIntuitive answerCorrect answer
Monty Hall: win by switching1/22/3
Bertrand’s box: other coin is gold1/22/3
Boy or girl: two boys, given at least one boy1/21/3
Birthday: shared birthday among 23 peopleabout 6%50.7%

Common mix-ups

  • The gambler’s fallacy. After five heads, the next flip of a fair coin is still 50–50. Streaks are more common than people expect, as the streak calculator shows.
  • European vs American roulette. The single zero gives a house edge of 2.70%; the extra double zero raises it to 5.26% on most bets.
  • Bertrand’s paradox has no single answer. “A random chord” is not well defined, and three natural methods give 1/3, 1/2 and 1/4.

Guides to read alongside

Common questions

Why is switching better in the Monty Hall problem?
Your first pick is right one time in three, and the host, who knows where the car is, always opens a goat door. Switching therefore wins whenever your first pick was wrong, which happens two times in three.
Are the coin flipper and dice roller truly random?
They use the browser’s cryptographic random number generator, which is unpredictable for all practical purposes. Over many flips or rolls the results settle close to the expected proportions, with the usual random scatter.
Why does the birthday paradox need only 23 people?
The question is whether any two people share a birthday, not whether someone shares yours. With 23 people there are 253 pairs, so the chance that every pair differs falls just below one half.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.