Experiments with only 2 outcomes are known as Bernoulli experiments. Examples are:
- Heads or tails in a coin toss
- Win or lose
- True or False
- Yes or No
- Binary code 0 or 1
The list is not limited to the ones shown above. Outcomes of any experiment can be divided into 2 categories<
- desired outcome
- outcomes other than the desired one
Let’s assume the probability of the desired outcome is p , the probability of seeing anything else will be 1-p because the total probability for outcomes is 1.
Probability of the desired outcome = p
Probability of not getting the desired outcome = 1-p
In the “relative frequency” interpretation of probability, the uncertainty of p in a Bernoulli experiment is proportional to
which peaks at p=0.5 (see the curve below). The uncertainty is highest when the odds are 50:50.
This is the explanation given by the experimental physicist Johann Summhammer . The example uses a physics experiment but you can apply the results to any Bernoulli experiment (coin flips, betting on the stock price direction, etc.)
Example: We have 2 detectors. We have done N trials and registered L clicks in detector 1. The relative frequency L/N is the sample estimate of the the true (unknown) probability ptrue that the particle hits detector 1. The random variable L/N is subject to a binomial distribution. Since this distribution has a finite standard deviation, it fulfills Chebyshev’s inequality
where the standard deviation is
this inequality means that, the probability that the ratio L/N will deviate from ptrue by more than k standard deviations is less than or equal to 1/k2. The arbitrary confidence parameter k only makes sense for k>1. The best estimate (sample estimate) for ptrue is pb=L/N.
With this definition, the interval, within which ptrue can be found with probability larger than (1-1/k2) , is obtained as
For sufficiently large N this reduces to the interval
which is interpreted as the uncertainty of the sample estimate of probability p from N trials.
Ignore the magnitude of the uncertainty and focus on the functional dependence which is given by
 Johannn Summhammer, “Invariants of Elementary Observation”, arXiv:quant-ph/0008098