Archive for urn models

a genuine riddle

Posted in Books, Kids, pictures with tags , , , , , on August 5, 2022 by xi'an

A riddle from The Riddler that was pure (if straightforward) logic rather than brute force compulation or mathematical modelling:

Four bags of many marbles are labelled R(ed), B(lue), G(green) and μ (mixed), except that all labels are wrong. Given the possibility to draw two balls, one at a time, from any bag, is it possible to select two monochromatic bags?

Bag μ draw is returning a color, R say, as it is a monochromatic bag. Drawing from another color bag, B say, will produce R or B, in which case it is μ, i.e., mixed (polychromatic), which means the other bags are monochromatic, or G. For this last case, bag B is either polychromatic, in which case bag G is made of blue marbles and bag R of green marbles, or monochromatic, in which case bag G is mixed and bag R is full of blue marbles, but monochromatic for either situation, hence to be chosen on top of bag μ.

a non-riddle

Posted in Books, Kids, R with tags , , on July 12, 2019 by xi'an

Unless I missed a point in the last riddle from the Riddler, there is very little to say about it:

Given N ocre balls, N aquamarine balls, and two urns, what is the optimal way to allocate the balls to the urns towards drawing an ocre ball with no urn being empty?

Both my reasoning and a two line exploration code led to having one urn with only one ocre ball (and no acquamarine ball) and all the other balls in the second urn.

odz<-function(n,m,t) 2*m/n+(t-2*m)/(t-n)
probz=matrix(0,trunc(N/2)-1,N-1)
for (n in 1:(N-1))
  for (m in 1:(trunc(N/2)-1))
    probz[m,n]=odz(n,m,N)

Frequency vs. probability

Posted in Statistics with tags , , , , , , , on May 6, 2011 by xi'an

Probabilities obtained by maximum entropy cannot be relevant to physical predictions because they have nothing to do with frequencies.” E.T. Jaynes, PT, p.366

A frequency is a factual property of the real world that we measure or estimate. The phrase `estimating a probability’ is just as much an incongruity as `assigning a frequency’. The fundamental, inescapable distinction between probability and frequency lies in this relativity principle: probabilities change when we change our state of knowledge, frequencies do not.” E.T. Jaynes, PT, p.292

A few days ago, I got the following email exchange with Jelle Wybe de Jong from The Netherlands:

Q. I have a question regarding your slides of your presentation of Jaynes’ Probability Theory. You used the [above second] quote: Do you agree with this statement? It seems to me that a lot of  ‘Bayesians’ still refer to ‘estimating’ probabilities. Does it make sense for example for a bank to estimate a probability of default for their loan portfolio? Or does it only make sense to estimate a default frequency and summarize the uncertainty (state of knowledge) through the posterior? Continue reading

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