Probability Confidence
Probability Confidence
Examination of Confidence as used in probability, and ways to optimize it’s selection
Introduction
Introduction
Example Problem
Example Problem
A casino offers you a gamble with a 1% chance of winning a try. How many tries will it take to win at least once?
Solution
Solution
For this example, I chose 95% confidence, a willingness to be wrong once in twenty
In[]:=
confidence=0.95;chanceToBeWrong=1-confidence;chanceOfSuccess=0.01;chanceOfFailure=1-chanceOfSuccess;tries=;Quantity[IntegerPart[tries+0.5],IndependentUnit["tries"]]
Log[chanceToBeWrong]
Log[chanceOfFailure]
Out[]=
But what about that decision to choose 95% confidence?
It was an arbitrary choice, governed by nothing more than a general rule of thumb. Can we do better?
It was an arbitrary choice, governed by nothing more than a general rule of thumb. Can we do better?
Research
Research
Confidence is a linear variable, a Real in the range 0 - 1.
It can be expressed in this example problem as confidence = 1 - ( -0.0099 * tries + 0.9999 )
If I choose confidence = 0.5, then I' m making a prediction of the future designed to be wrong half the time
If I choose confidence = 1, then it would take infinite tries
So the range of admissible values for confidence appears to be (0.5, ..., 1) excluding end points
It can be expressed in this example problem as confidence = 1 - ( -0.0099 * tries + 0.9999 )
If I choose confidence = 0.5, then I' m making a prediction of the future designed to be wrong half the time
If I choose confidence = 1, then it would take infinite tries
So the range of admissible values for confidence appears to be (0.5, ..., 1) excluding end points
You could construct a confidence interval with less than 50% confidence.
Binomial Distribution
Binomial Distribution
The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:1-p
f(k,n,p)=Pr(k;n,p)=Pr(X=k)=
(
n |
k |
k
p
n-k
)
for k = 0, 1, 2, ..., n, where
=
n |
k |
n!
k!(n-k)!