Probability in simple words
Probability is how likely something is to happen, as a number from 0 to 1. 0 means it can never happen, 1 means it is certain, and 0.5 means it is an even chance, like a coin landing on heads. You can also write it as a percent: 0.5 is 50%.
When two things do not affect each other (they are independent), the chance of both happening is the two chances multiplied. A coin landing heads (0.5) and a die showing 6 (1/6 ≈ 0.1667) together is 0.5 × 0.1667 ≈ 0.083, about 8%.
The chance of one or the other (or both) is the two chances added, minus the overlap so it is not counted twice: 0.5 + 0.1667 − 0.083 ≈ 0.583, about 58%.
The chance that something does not happen is 1 minus the chance that it does. The die not showing 6 is 1 − 1/6 = 5/6, about 83%.
Two events: independence versus exclusion
For independent events, P(A and B) = P(A)P(B). For mutually exclusive events, P(A and B) = 0. In both cases, P(A or B) = P(A) + P(B) − P(A and B). Independence and mutual exclusion are different assumptions, not interchangeable options.
For P(A) = 0.5 and P(B) = 0.3 with independence, the chance of both is 0.15, either is 0.65, and neither is 0.35. General dependent events require additional information and are not modeled here.
Exactly, at least and at most
For n independent trials with constant success probability p, P(X = k) = C(n, k)p^k(1 − p)^(n − k). Add terms from k through n for “at least k”, or from 0 through k for “at most k”. Expected successes equal np, and standard deviation is √[np(1 − p)].
In ten fair coin flips, exactly three heads has probability 120 ÷ 1,024 = 0.1171875. At most three heads is 0.171875; at least three is 0.9453125. The chance of at least one head is 1 − (0.5)^10 = 0.9990234375.
Input and model limits
Enter a probability as a decimal from 0 to 1 or with an explicit percent sign, such as 25%. A bare 25 is rejected. Trials must be whole numbers from 1 to 1,000 and successes must be whole numbers between zero and the trial count.
Repeated-trial models assume the same success probability on each independent trial. Sampling without replacement generally violates that assumption. Very small displayed probabilities can round to zero; logarithmic computation reduces loss from intermediate powers but cannot represent arbitrarily tiny values.