Based on Babich (1992). Suppose that each week eachof 300 families buys a gallon of orange juice fromcompany A, B, or C. Let pA denote the probability thata gallon produced by company A is of unsatisfactoryquality, and define pB and pC similarly for companiesB and C. If the last gallon of juice purchased by a family is satisfactory, the next week they will purchase agallon of juice from the same company. If the last gallon of juice purchased by a family is not satisfactory,the family will purchase a gallon from a competitor.Consider a week in which A families have purchasedjuice A, B families have purchased juice B, and Cfamilies have purchased juice C. Assume that familiesthat switch brands during a period are allocated tothe remaining brands in a manner that is proportionalto the current market shares of the other brands. Forexample, if a customer switches from brand A, thereis probability B/(B 1 C) that he will switch to brandB and probability C/(B 1 C) that he will switch tobrand C. Suppose that the market is currently dividedequally: 10,000 families for each of the three brands.a. After a year, what will the market share for each firmbe? Assume pA 5 0.10, pB 5 0.15, and pC 5 0.20.(Hint: You will need to use the RISKBINOMIALfunction to see how many people switch from Aand then use the RISKBINOMIAL function againto see how many switch from A to B and fromA to C. However, if your model requires moreRISKBINOMIAL functions than the number allowed in the academic version of @RISK, rememberthat you can instead use the CRITBINOM functionto generate binomially distributed random numbers.This takes the form 5CRITBINOM(ntrials,psuccess,RAND()).)b. Suppose a 1% increase in market share is worth$10,000 per week to company A. Company Abelieves that for a cost of $1 million per year it cancut the percentage of unsatisfactory juice cartonsin half. Is this worthwhile? (Use the same values ofpA, pB, and pC as in part a.)
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Based on Babich (1992). Suppose that each week each
of 300 families buys a gallon of orange juice from
company A, B, or C. Let pA denote the probability that
a gallon produced by company A is of unsatisfactory
quality, and define pB and pC similarly for companies
B and C. If the last gallon of juice purchased by a family is satisfactory, the next week they will purchase a
gallon of juice from the same company. If the last gallon of juice purchased by a family is not satisfactory,
the family will purchase a gallon from a competitor.
Consider a week in which A families have purchased
juice A, B families have purchased juice B, and C
families have purchased juice C. Assume that families
that switch brands during a period are allocated to
the remaining brands in a manner that is proportional
to the current market shares of the other brands. For
example, if a customer switches from brand A, there
is probability B/(B 1 C) that he will switch to brand
B and probability C/(B 1 C) that he will switch to
brand C. Suppose that the market is currently divided
equally: 10,000 families for each of the three brands.
a. After a year, what will the market share for each firm
be? Assume pA 5 0.10, pB 5 0.15, and pC 5 0.20.
(Hint: You will need to use the RISKBINOMIAL
function to see how many people switch from A
and then use the RISKBINOMIAL function again
to see how many switch from A to B and from
A to C. However, if your model requires more
RISKBINOMIAL
that you can instead use the CRITBINOM function
to generate binomially distributed random numbers.
This takes the form 5CRITBINOM(ntrials,
psuccess,RAND()).)
b. Suppose a 1% increase in market share is worth
$10,000 per week to company A. Company A
believes that for a cost of $1 million per year it can
cut the percentage of unsatisfactory juice cartons
in half. Is this worthwhile? (Use the same values of
pA, pB, and pC as in part a.)
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