Probability Calculator

Basic & combined event probability

A single event's probability is favorable outcomes divided by total outcomes. For two events, P(A or B) = P(A) + P(B) − P(A and B) — subtracting the overlap so it isn't double-counted. Conditional probability, P(A given B), is P(A and B) ÷ P(B): the chance of A once you already know B happened. Two events are independent when P(A and B) exactly equals P(A) × P(B); this calculator checks that for you automatically.

The overlap can't be just any number. It can't be bigger than the smaller of the two events, and it can't be smaller than P(A) + P(B) − 100%. If two events each have an 80% chance, at least 60% of the time both must happen, because there isn't room for them to avoid each other. Enter an overlap outside that range and the calculator tells you the valid range instead of returning an impossible answer such as 150%.

Dice, coin and card examples

Games of chance are the easiest place to check your intuition, because every outcome is equally likely and you can count them. These assume fair dice and coins and a standard, well-shuffled 52-card deck:

Two habits cover most of these. For "at least one" questions, work out the chance of none and subtract it from 1. For draws without replacement, the second fraction changes because the deck has shrunk: 3 aces left out of 51 cards. To see how real results wander around these numbers, roll a few hundred virtual dice with the random number generator and compare the counts.

Permutations vs. combinations

Counting outcomes is half of most probability problems. Use a permutation when order matters and a combination when it doesn't. Choosing a president, treasurer and secretary from 5 people is a permutation: 5P3 = 5! ÷ (5 − 3)! = 5 × 4 × 3 = 60 ways. Choosing any 3-person committee from the same 5 is a combination: 5C3 = 5! ÷ (3! × 2!) = 10, because each committee can be ordered 3! = 6 ways and those all count as one.

Lotteries are combinations. Picking 6 numbers from 49 gives 49C6 = 13,983,816 possible tickets, so one ticket's chance of matching all six is 1 in 13,983,816, about 0.0000072%. The same nCr term, "n choose k", is the first factor in the binomial formula below. You can check any factorial or power by hand on the scientific calculator.

Binomial probability

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same success probability — coin flips, pass/fail tests, yes/no survey responses. P(X = k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) is "n choose k," the number of ways to pick k successes from n trials. The mean number of successes is n × p, and the standard deviation is √(n × p × (1−p)).

Example: a multiple-choice quiz has 10 questions with 4 options each, and you guess every one. With n = 10 and p = 25%, the chance of exactly 3 right is C(10,3) × 0.25³ × 0.75⁷ = 120 × 0.015625 × 0.1335 ≈ 25.03%, and the expected score is 10 × 0.25 = 2.5. The calculator works in logarithms internally, so it stays accurate for up to 10,000 trials and still reports very small tail probabilities (such as 95 or more heads in 100 flips) instead of rounding them to zero.

Implied probability from betting odds

Betting odds encode an implied probability of an outcome. For American odds: a positive number's implied probability is 100 ÷ (odds + 100); a negative number's is |odds| ÷ (|odds| + 100). For decimal odds, it's simply 1 ÷ decimal odds. This calculator converts between American and decimal odds automatically, alongside the implied probability — note that real sportsbook odds also bake in a margin ("vig"), so the true probability is usually slightly lower than what's implied.

You can see the margin by adding up both sides of a market. A game priced at −110 on each team implies 52.38% + 52.38% = 104.76%; the 4.76 points above 100% are the bookmaker's edge.

Z-scores and the normal distribution

A z-score measures how many standard deviations a value is from the mean of a normal distribution. This calculator converts a z-score into a probability using the standard normal cumulative distribution function — P(Z ≤ z) is the area under the curve to the left of z. The familiar z = 1.96 corresponds to the 95% confidence interval boundary used throughout statistics.

The short z-table below lists the values that come up most often. For a negative z, use symmetry: P(Z ≤ −z) = 1 − P(Z ≤ z), so P(Z ≤ −1) = 1 − 0.8413 = 0.1587.

To get a z-score from raw data, subtract the mean and divide by the standard deviation; the standard deviation calculator gives you both from a list of numbers.