Probability Calculator

Probability in plain numbers

Probability is a number between 0 and 1 that says how often an outcome happens over many repeats. Zero means it cannot happen, one means it must, and 0.5 means it happens about half the time. Multiply by 100 to read the same number as a percent.

The chance of rolling a 4 on a fair six-sided die is 1/6 = 0.1667, or 16.7 percent. The chance of rolling something other than a 4 is 1 - 0.1667 = 0.8333, or 83.3 percent. That is the complement rule, and it is the fastest check on any answer: the probability of an event plus the probability of its opposite has to equal 1.

The NIST tutorial on probability states the same rules in formal terms: a probability sits between 0 and 1, and for two events the addition rule subtracts the overlap because it would otherwise be counted twice.

The addition rule for either event

When you want the chance that A happens, B happens, or both happen, add the two probabilities and subtract the overlap:

P(A or B) = P(A) + P(B) - P(A and B)

One die gives a clean example. Let A be "2 or less", which covers 1 and 2, so P(A) = 2/6. Let B be "even", which covers 2, 4 and 6, so P(B) = 3/6. The overlap is rolling a 2, which is 1/6. The union is 2/6 + 3/6 - 1/6 = 4/6 = 66.7 percent.

Forget the subtraction and you get 5/6, which overcounts the 2. The overlap is not an edge case; it is the part of the answer most people double count. The Richland Community College statistics notes walk through the same rule with the same warning about the double count.

Independent, mutually exclusive, and neither

Two events are independent when one result does not change the other. Two coin flips are independent: the second flip is 50 percent no matter what the first one did. For independent events, multiply:

P(A and B) = P(A) x P(B)

Two heads in a row: 0.5 x 0.5 = 0.25, or 25 percent. A basketball player who makes 80 percent of free throws, taking two independent shots, makes both 0.8 x 0.8 = 0.64 of the time, or 64 percent.

Mutually exclusive events cannot happen in the same trial. Rolling a 2 and rolling a 5 are mutually exclusive, so their overlap is zero and the addition rule collapses to P(A) + P(B) = 1/6 + 1/6 = 1/3.

The two ideas are often confused. If both events have a probability above zero, they cannot be both independent and mutually exclusive: independence says the overlap is P(A) x P(B), which is positive, while mutual exclusivity says it is zero.

Conditional probability and the base rate trap

Conditional probability asks how likely A is once you know B happened:

P(A | B) = P(A and B) / P(B)

In a 52-card deck, 12 cards are face cards and 4 of those are kings. The chance of a king, given a face card, is 4/12 = 33.3 percent. The condition shrinks the sample space from 52 cards to 12.

Medical screening shows why the base rate matters. Suppose 1 percent of a population has a condition, a test catches 95 percent of true cases, and 5 percent of healthy people test positive anyway. In 10,000 people, 100 have the condition and 95 of them test positive. Of the 9,900 healthy people, 495 test positive. A positive result therefore comes from 95 true cases out of 590 positives:

95 / 590 = 0.161, so the chance of having the condition after a positive test is 16.1 percent, not 95 percent. The false positives outnumber the true ones because the healthy group is 99 times larger. This is the base rate trap, and it is the most common error in reading any test result.

Odds are not probability

Odds compare the chance of an event with the chance against it. Probability compares the event with all possible outcomes. The conversion is:

odds = p / (1 - p)

A 25 percent chance gives odds of 0.25 / 0.75 = 1/3, written 1 to 3. A 90 percent chance gives 0.9 / 0.1 = 9, written 9 to 1. To go the other way, p = odds / (1 + odds), so 1 to 3 becomes 0.3333 / 1.3333 = 0.25.

The two scales drift apart at the extremes, which is where most misreading happens. A 1 percent chance is 1 to 99 in odds, while a 50 percent chance is 1 to 1. If a headline says "odds doubled", the probability did not double unless it started at 50 percent.

Repeated trials and the binomial formula

For a fixed number of independent trials with the same success probability, the chance of exactly k successes is:

P(X = k) = C(n, k) x p^k x (1 - p)^(n - k)

C(n, k) counts the orders in which the successes can appear. For 10 coin flips and exactly 5 heads, C(10, 5) = 252, and 252 / 1,024 = 0.246, or 24.6 percent. The same count answers a scheduling question: C(10, 3) = 120 ways to choose 3 days out of 10, while P(10, 3) = 720 if the order of those days matters.

The distribution is symmetric around 5 heads, and the two ends are thin: 0 heads and 10 heads are 0.1 percent each. The mean is n x p = 10 x 0.5 = 5, and the standard deviation is the square root of n x p x (1 - p), which is 1.58 heads. Most results land within one standard deviation of the mean, so 4 to 6 heads covers about 66 percent of runs.

At least one success

"At least one" is easier through the complement. The chance of no success in n independent tries is (1 - p)^n, so:

P(at least one) = 1 - (1 - p)^n

Roll a die four times and the chance of at least one 6 is 1 - (5/6)^4 = 1 - 0.4823 = 0.5177, or 51.8 percent. Ten attempts at a 20 percent chance gives 1 - 0.8^10 = 1 - 0.1074 = 0.8926, or 89.3 percent.

The curve flattens: at 20 percent per try, 10 tries reach 89.3 percent, but 20 tries only reach 98.8 percent. Going from 10 to 20 tries buys less than 10 points, while the first 10 tries bought 89.

Where probability estimates go wrong

Most bad probability statements fail on one of five points: the events are not independent, the overlap is ignored, the base rate is dropped, the odds are read as a probability, or the sample is too small to say anything.

Two habits catch most of these. First, write the sample space down before computing: how many equally likely outcomes exist, and how many match the event. Second, sanity check the result against 0 and 1. Any answer outside that range is an arithmetic error, and an answer of exactly 0 or 1 usually means an assumption was too strong.

For anything with money or health attached, treat the output as a model, not a fact. The formulas are exact; the inputs are estimates.