15 8. How Randomness Rules Our Lives Albert Einstein struggled to accept the fac

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15 8. How Randomness Rules Our Lives Albert Einstein struggled to accept the fact that the quantum world is probabilistic rather than deterministic and famously stated, "God doesn't play dice with the universe." This is more than just a nice metaphor. In Leonard Mlodinow's book on probability theory, The Drunkard's Walk: Randomness Rules Our Lives, he examines the 1961 competition between Mickey Mantle and How Roger Maris, who were engaged in an historic race to tie or break Babe Ruth's beloved record in 1927 of 60 home runs in one year. He considers "all players with the talent of Maris and the entire venty-year period from Ruth's record to the start of the "steroid era." Mlodinow asks, "what are the odds that some player at some time would have matched or broken Ruth's record by chance alone?" He turns to an August 1961 issue in Life magazine, which includes a discusesion of this probability theory. Mlodinow concludes that it turns out to be a little greater than 50%. He also remarks, "detailed analysis of baseball and other sports by scientists as eminent as the late Stephen Jay Gould and the Nobel Laureate E.M. Purcell show that coin-tossing models like the one I've described match very closely the actual performance of both players and teams. In this question, we develop an exercise in your textbook and show how it can used to model competition in sports. Suppose an intramural hockey league has four teams that maintain standings with points throughout the season. If team X is higher in the standings than team Y, and team Y higher in the standings that team Z, then of course team X will be higher than team Z. This For instance suppose team X has z points, team Y has y points, and team Z has z points. If > y and y > z then z > z. In other words, the relation "higher in the standings" is transitive. If two teams are going to play one another, then, sports analysts often look at how they match up against one another. This may be a better indicator of which team is favored to win in a game than their position in the standings. However, the relation "favored over" is not necessarily transitive! For instance, based on individual match-ups, Pittsburgh may be favored over New York, and New York favored over Philadelphia, but Pittsburgh may not be favored over Philadelphia. The psychology of sport is a fascinating subject, and rivalries in particular play a major role in the entertainment value Suppose the four teams in the league are A, B,C and D, and that each team is represented by a standard die with six faces. The dice are loaded to reflect the results of their previous matches as shown in A 3, 3, 4, 4, 8, 8 the accompanying table. A match between two teams is simulated by B 1,1, 5,5,9, 9 rolling their dice. If Xs die turns up higher than Ys, then we say X beats" Y. Moreover, we say "X is favored over Y" if the probability that X beats Y is greater than the probability that Y beats X. In D 3,3, 5, 5, 7,7 summary, we write aces C 2, 2, 6, 6, 7, 7 If we write X Y to denote that "X is favored over Y," then 10 X~Y exactly when P(X beats Y)> P(Y beats X) (t) a) When two standard dice are rolled, the sample space has size 36. However, in this case each die has only 3 equally likely outcomes so this amounts to a sample space of what reduced size? after which the ed Som owner sold Babe Ruth to the New York Yankees, fimilariy, the long-standing

Explanation / Answer

a. Each dice has 3 possibilities. SO, total number of outcomes in roll of 2 dice is 3*3 = 9

b. P(C beats B) = (2,1) , (6,1), (6,5), (7,1), (7,5), i.e. when C gets 2 and B gets 1 and so on = 5/9

So, C is favored over B

P(B beats A) = 5/9. So, B > A

P(C beats A) = 4/9 So, A>C.

P(A beats D) = 4/9 = P(D beats A)

P(B beats D) = 4/9 = P(D beats B)

P(C beats D) = 4/9 = P(D beats C)

c. The law of trichotomy is followed as we can have only 3 cases : c=y or x>y or x<y and they are exclusive cases

The law of transitivity does not hold as we had that C>B, B>A but A>C

So, it is not order relation as transitivity is not followed

d. Again for A, B, C and D we get the same results as in part c above.

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