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.