Probability Calculator

How to use the probability calculator

Choose one of three modes:

• Two independent events: enter P(A) and P(B) to get the chance of both, at least one, exactly one or neither happening, plus each complement.
• At least one success in n tries: enter the chance of success on each try and the number of tries. You also see how many tries you need for a 50%, 90% or 99% chance.
• Exactly k successes in n tries: the binomial distribution, with cumulative probabilities (at most k, at least k) and a full table of outcomes for up to 60 tries.

Probabilities can be typed as 0.25, 25% or 1/4. A plain number between 1 and 100, such as 30, is read as a percentage, and the result tells you when that happens.

Probability formulas for two independent events

Two events are independent when one happening does not change the chance of the other, such as flipping a coin and rolling a die. With P(A) = 0.5 (heads) and P(B) = 1/6 (rolling a six):

What you want	Formula	Heads and a six
Both A and B	P(A) × P(B)	0.083333 (1 in 12)
A or B (at least one)	P(A) + P(B) − P(A) × P(B)	0.583333
Exactly one (A xor B)	P(A) + P(B) − 2 × P(A) × P(B)	0.5
Neither	(1 − P(A)) × (1 − P(B))	0.416667
Not A	1 − P(A)	0.5
Not B	1 − P(B)	0.833333

The “or” formula subtracts the overlap because adding P(A) and P(B) counts the case where both happen twice. Note that “or” plus “neither” always makes 1: 0.583333 + 0.416667 = 1.

Worked example: at least one six in four rolls

The French gambler known as the Chevalier de Méré asked why betting on at least one six in four rolls of a die seemed to win, while betting on at least one double six in 24 rolls of two dice seemed to lose. His puzzle is one of those discussed in the 1654 letters between Blaise Pascal and Pierre de Fermat that laid the foundations of probability theory.

The trick is to look at the chance of failing every time:

1. The chance of not rolling a six on one roll is 5/6 = 0.833333.
2. The rolls are independent, so the chance of no six in four rolls is 0.833333⁴ = 0.482253.
3. The chance of at least one six is 1 − 0.482253 = 0.517747, just over 51.77%. A small edge for the bettor.

For a double six in 24 rolls, the chance per roll is 1/36, and 1 − (35/36)²⁴ = 0.491404, just under 50%. De Méré’s instinct was right, even though the two bets look alike.

Exactly k successes: the binomial formula

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ

• n = number of independent tries
• k = number of successes you want
• p = chance of success on each try
• C(n, k) = n! ÷ (k! × (n − k)!), the number of ways to choose which tries succeed

Exactly 2 sixes in 10 rolls of a die: C(10, 2) = 45, (1/6)² = 0.027778 and (5/6)⁸ = 0.232568, so P = 45 × 0.027778 × 0.232568 = 0.29071, about 29%. The chance of at most 2 sixes is 0.775227, and of at least 2 sixes is 0.515483. The mean number of sixes is n × p = 1.6667.

How many tries until success is likely?

Solving 1 − (1 − p)ⁿ ≥ target for n gives n = ln(1 − target) ÷ ln(1 − p), rounded up. For an item with a 1% drop rate in a game, or any 1-in-100 chance:

Target chance	Tries needed at p = 1%	Tries needed at p = 1/6
50%	69	4
90%	230	13
99%	459	26

Even after 100 tries at 1%, there is still a 36.6% chance of no success at all. Bad luck streaks are more common than intuition suggests.

Common probability mistakes

• Adding chances across tries. Six rolls at 1/6 each is not a 100% chance of a six. It is 1 − (5/6)⁶ = 66.51%.
• The gambler’s fallacy. After five tails in a row, a fair coin is still 50% heads on the next flip. Independent tries have no memory.
• Treating dependent events as independent. Drawing two aces from a deck without replacement changes the odds on the second draw, so the simple product rule does not apply.
• Mixing percentages and decimals. 30% is 0.3, not 30. Multiplying 30 × 50 instead of 0.3 × 0.5 gives nonsense.

To draw fair random numbers for a game or experiment, use the random number generator. For probabilities under a bell curve, such as test scores or measurements, use the z-score calculator. The fraction calculator is handy for keeping dice and card probabilities exact.