Bernoulli Trial Calculator

How it is calculated

A Bernoulli trial has two outcomes: success with probability p and failure with probability 1 − p. Counting success as 1 and failure as 0, the mean is E = p, the variance is σ² = p·(1 − p) and the standard deviation is σ = √(p·(1 − p)). Enter p as a probability from 0 to 1 inclusive, so 25 % is 0.25, not 25; anything outside that range, or a blank or non-numeric entry, gives no result. Outputs show 2 decimals, but a small non-zero value keeps extra digits so that p = 0.001 reads a mean of 0.0010 and a variance of 0.00100 rather than 0.00. The variance is largest at p = 0.5, where it is 0.25.

Common mistakes

• Entering a percentage such as 30 instead of 0.3. Only 0 to 1 is accepted, so 30 shows no result and the 30 % case is 0.3.
• Reading the mean as a count of successes. For one trial it is the probability itself; for n trials you multiply by n.
• Using the result for several trials. The standard deviation of a total of n trials is √(n·p·(1 − p)), not n times this value.
• Expecting the variance to grow with p. It peaks at p = 0.5 and falls toward 0 near both ends, so p = 0.1 and p = 0.9 have equal variance.
• Reading the standard deviation as tiny for a rare event. For p = 0.001 the variance is 0.00100 but the standard deviation is about 0.03, far larger than p, because the square root of a small number is much bigger than the number itself.