3. A magical four-sided die is rolled twice. Let S be the sum of the resultsof the two rolls. We are told that the probability that S = k is proportional to k, for k = 2, 3, . . . , 8, and that all possible ways that a given sum k can arise are equallylikely. Construct an appropriate probabilistic model and find the probability of gettingdoubles.

P(S=k) = (1/35)k

8/21

Step-by-step explanation:

P(S=k) is proportional to k

P(S=k) = ck

Total probability is 1

k = 2,3 8

Sum of all probabilities =

2c + 3c + 4c + 5c + 6c + 7c + 8c = 1

35c = 1

c = 1/35

P(S=k) = (1/35)k, where k = 2,3,...8

P(1&1):

P(S=2) = 2/35

P(2&2):

P(S=4)/3 = (1/3)(4/35) = 4/105

Because for S=4: (1,3) (2,2) (3,1) and all are equally likely

P(3&3):

P(S=6)/3 = (1/3)(6/35) = 2/35

Because for S=6: (2,4) (3,3) (4,2)

P(4&4):

P(S=8) = 8/35

P(getting doubles = 2/35 + 4/105 + 2/35 + 8/35 = 40/105 = 8/21

7200

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