Solution to “The Logical Flockmistress Problem
January / February, 2012 Issue
Elaine Bryant, 14 January 2012

Statement of the Problem:
Now the Logical Flockmistress had a lot of different kinds of sheep and goats.  She had twelve Wiltshire Horn, two Dorpers, seventeen St. Croix, fifteen Katahdins, six Black Hawaiians, thirteen Barbados Blackbellies, four Pygora goats, and four Angora goats.
One day when the Logical Flockmistress and her husband were sitting on the front porch talking about the weather, a city man came up to them and asked the following question:
"Would it be alright if I took a picture of one of your sheep?"
"Help yourself," declared the Logical Flockmistress.  "All my sheep and goats are out in the pasture there."  And so the man went off to the pasture to take his photograph.
"Now what do you suppose the odds are that he will take a picture of a St. Croix?" the Logical Flockmistress asked her husband.
"I haven't the faintest idea," he replied, discouraged because once again he was being asked to solve another blessed math problem.
"Wait until the kids find out that you don't know how to figure out something so simple!", his wife said.
"Well alright," said the husband, and he got to work and figured out the odds.

Problem Assessment and Initial Solution
    The Logical Flockmistress' husband began to think about the problem and said to his wife, "You know, that city fella sure was all dressed up, I wonder if he has ever been in the country before?"  "For all we know," he continued, "he may not even know a sheep from a goat!"  As he said this, he realized that the odds of taking a picture of a St. Croix were different depending on whether the man knew that difference or not.
    The Logical Flockmistress' husband felt the urge to define and assign some variables, so initially, he proceeded to define  as the total number of sheep in the pasture and  {breed = Wiltshire Horn, Dorpers, St. Croix, Katahdins, Black Hawaiians, or Barbados Blackbellies} being the quantity of sheep of a specific breed in the overall population, ; and  as the total number of goats in the pasture and  {breed = Pygora goats and Angora goats} being the quantity of goats of a specific breed in the overall population, .  He computed these totals as shown in  and  below.
     
        
Hence the total number of animals in the pasture was , or 73.
    As he contemplated the city man approaching the pasture, he realized the man's selection of an animal for a photograph might be influenced by the distribution of animals across the pasture (i.e., he would be more likely to select an animal closer to him than one on the other side of the pasture).  Furthermore, he realized that the man's decision might also be influenced by animal clustering (i.e. he would be more likely to photograph a lone animal than one that he would have to try to separate from the cluster.  Finally, since he was not at all sure that the city man could differentiate a sheep from a goat, he might actually select a goat for the subject of his sheep photograph.
    Thus, the Logical Flockmistress' husband concluded that the probability of selecting any one individual animal was described by the joint probability density function of the individual density functions of (i) selecting an individual animal independent of location, (ii) selecting an animal based on instantaneous field position and proximity to the man's approach, (iii) selecting an animal based upon the probability of instantaneous animal clustering, and (iv) the man's ability to differentiate a sheep from a goat.  The instantaneous characterization of the two density functions for (ii) and (iii) above recognizes the time-dependency of these distributions, and that fact that the density functions of interest are only the ones realized at the "instant" the man makes his animal selection for the photograph.
  

 The Logical Flockmistress' husband decided to consider each of these interacting density functions in turn, as follows:
(i) Individual Animal Selection (IAS): The independent selection of one animal from a population of animals is clearly an example of a single Bernoulli trial under a binomial distribution.  Under this type of distribution independent selections are equiprobable as traditionally exemplified by flipping a fair coin or rolling a fair die.  In this case, for a population of  items, the probability of selecting any one individual item at random is simply the reciprocal of the population size, or ; consider the analogy that the probability of realizing a "heads" from the toss of a fair coin is  or realizing a , on the roll of a fair die is .
(ii) Instantaneous Field Position (IFP): The instantaneous position of an animal in the pasture describes the probability of an individual animal being in a given position at a given time.  This position, taken independently of animal clustering, is also equiprobable as any given animal can be anywhere in the field at any point in time (this ignores the possibility of an individual animal having a "favorite" spot in the pasture or two animals trying to occupy the same spot in the pasture simultaneously).  Therefore, since this is an instantaneous sampling of the distribution, not a sampling of the average distribution, the Logical Flockmistress' husband concluded the IFP really had no bearing on the probability of the city man selecting a particular animal since all IPF distributions are equiprobable at the time of his selection.
(iii) Instantaneous Animal Clustering (IAC): The analysis of the instantaneous clustering of animals follows a similar argument as the IFP distribution above.  While animal clustering is a characteristic of many animals such as sheep and goats and as such, it has a non-zero probability of occurring, this only has bearing on the average or central tendency of the distribution.  From an instantaneous sampling of the distribution point of view, all animals are equiprobable as to whether they are in a cluster or not (note: this is different than the probability of clustering).  This statement assumes that all breeds of animals have the same tendency to cluster, and all animals across those breeds have the same propensity to cluster (i.e., there are not some animals who are in a cluster all the time, and other animals that rarely cluster).
(iv) Animal Discrimination Index (ADI): The ADI is a discrete bi-valued index (either 0 or 1) that is based on , the probability that the city man can accurately discriminate a goat from a sheep.  The definition of the ADI is shown in  below.  
        
Note that even though this index can only assume two discrete values, the probability of which of those two values it assumes is governed by the distribution of .  Since the Logical Flockmistress' husband had no way of knowing the value of the ADI, he realized he could proceed in two alternate ways; (1) compute the expected value of ADI based on the distribution of , or (2) he could compute two different answers, one for ADI = 0 and the other for ADI = 1.  He decided to proceed with the second approach.
    Having essentially dismissed the impact of the instantaneous probability distribution of both field position and animal clustering, the Logical Flockmistress' husband was ready to proceed with the solution to his wife's question.  He assumed that based on the discussion above, the city man's selection of an animal to photograph was solely based upon the IAS distribution and the value of the ADI.  For the IAS, he had already determined that the animal selection was a Bernoulli trial based on an underlying binomial distribution.  Hence for a population of size , the probability of randomly picking any one individual within that population is .  If individuals within the population share some common characteristic, such as color or breed, then the probability of picking an individual from the population with that characteristic is simply . 

The Logical Flockmistress' husband was now able to apply this general result to a specific solution to the question as follows.
Case 1: The city man cannot differentiate sheep from goats (ADI=0)
Since for this case, it is assumed the city man cannot differentiate sheep from goats, then the relevant population is the set of all the animals (both sheep and goats) in the pasture, or .  Furthermore, the question that the Logical Flockmistress had asked was related the St. Croix breed of sheep; thus the characteristic being sampled from the underlying distribution is the sheep breed of St. Croix of which there are a total of  in the pasture.  Thus, the probability of the city man selecting a St. Croix for his photograph for this case is shown in  below.
        
Case 2: The city man can differentiate sheep from goats (ADI=1)
Since for this case, it is assumed the city man can differentiate sheep from goats, then the relevant population is limited to the sheep in the pasture, or .  The characteristic being sampled is still the sheep breed of St. Croix of which there are still  in the pasture.  Thus, the probability of the city man selecting a St. Croix for his photograph for this case is shown in  below.
        
    "So," the Logical Flockmistress' husband said to his wife, "if that city fella can tell a sheep from a goat, then the odds that he will photograph one of your St. Croix sheep is .  On the other hand, if he doesn't know the difference between a sheep and a goat, the odds of him getting a St. Croix is ."  They laughed thinking about the possibility of the city man departing with a photograph of a Pygora sheep!  "If he did that and he found out later," said the Logical Flockmistress, "that would really get his goat!"  They chuckled some more.

Problem Extension  
After they had a good laugh, the Logical Flockmistress' husband asked his wife, "What if he had wanted to take more than one photograph, what are the odds that he would take them only of St. Croix?"  A good question said his wife.  "Let's do a parametric study to determine that as a function of the number of photographs he would take.  We will have to assume that he would have enough sense not to take a photograph of the same animal twice though."
    They began to work on the problem, and they defined  to be the number of pictures that they assumed the city man would take.  They realized that subject each photograph would be independently selected, and hence be a series of Bernoulli trials.  Furthermore, since this was sampling without replacement, they knew that as each photograph was taken, the available population is decremented by one prior to the next sampling event.  Based on these groundrules, the couple formulated the problem as shown in  and  below.
Case 1: The city man cannot differentiate sheep from goats (ADI=0)
Once again here the total available population consists of both the sheep and the goats, .  The probability, therefore of selecting the jth St. Croix in a row for the photograph would be the probability of selecting (j-1) St. Croix in a row times , the ratio of the remaining St. Croix sheep to the remaining population.  In general, the probability for  photographs of a St. Croix in a row are shown in  below.
        
Case 2: The city man can differentiate sheep from goats (ADI=1)
Once again here the total available population consists of only the sheep, .  Similar to above, the probability for  photographs of a St. Croix in a row are shown in  below.
        
As expected, both these compounded probabilities rapidly approach zero, as it is intuitive that the probability of randomly selecting a series of only St. Croix sheep to photograph becomes very unlikely as the number of photographs increase.  Equations  and  were then solved numerically as a function of the number of photographs, .