Uncertainty is Highest at 50:50 Odds

Bernoulli Experiments

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:

  1. desired outcome
  2. 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

Uncertainty

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.

.

Explanation

This is the explanation given by the experimental physicist Johann Summhammer [1]. 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

where f=1/(1+k2/N).

For sufficiently large N this reduces to the interval

with

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

.

References

[1] Johannn Summhammer, “Invariants of Elementary Observation”, arXiv:quant-ph/0008098

 

 

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About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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