Question

Four beer drinkers (say I, II, III and IV) are to rank three different brands of beer (say A, B, and C) in a blindfold test. Each drinker ranks the four beers as 1 (for the beer that he or she liked best), 2 (for the next best) or 3.Assume that the drinkers cannot discriminate between the beers so that each assignment of ranks to the beers is equally likely. After all the beers are ranked by all three drinkers, the ranks of each brand of beer are summed.

a What is the probability that beer A will receive a total rank of 4 ?

b What is the probability that some beer will receive a total rank of 4 ?

c What is the probability that some beer will receive a total rank of 5 or less ?

Answer 1

a. Let R: total rank, P(R= 4 for A) = P(each drinker gives rank 1 to A) now, P(drinker gives rank 1 to A) = 2/6 = 1/3 so P(R = 4 for A) = (1/3)4 = 1/81 6. P(R = 4 for some beer) = P(R= 4 for A) + P(R= 4 for B) + P(R= 4 for C) Now, P(R 4 for A) = P(R = 4 for B) = P(R = 4 for C) = (1/3)4 = 1/81 so P(R 4 for some beer) = 3 * (1/81) = 1/27 Note it is impossible for two beers to simultaneously recieve total rank of 4.

$c.\ P(R\le 5\ for\ some)=P(R=4\ for\ some)+P(R=5\ for\ some) , \\ note\ R=4\ and\ R=5\ for\ some\ , are\ mutually\ exclusive\ events. \\ also\ note\ lowest\ possible\ rank\ is\ 4. \\ so\ P(R=5\ for\ some)=P(for\ some\ beer,\ 3\ of\ drinker\ gives\ rank1\ and\\ fourth\ drinker\ gives\ rank\ 2. )=^{4}C_{3}(1/3)^{3}(1/3)=4/81 \\ already\ from\ b, \ we\ know\ P(R=4\ for\ some)=1/27 \\ so\ P(R\le 5\ for\ some\ beer)=(1/27) + (4/81) = 7/81$