B. You have a fair five sided dice, the sides numbered by 1-5. You roll the dice

Question

B. You have a fair five sided dice, the sides numbered by 1-5. You roll the dice twice. (a) Let event A be ”the total of two rolls is 10”, event B be ”at least one roll results in 5”, and event C be ”at least one roll results in 1”. Are events A and B independent? Are events A and C independent? (b) Let event D be ”the total of the two rolls is 7”, event E be ”the difference between the two rolls is exactly 1”, and event F be ”the second roll is larger than the first roll”. Are events E and F conditionally independent given D?

Explanation / Answer

The die is 5 sided

Equal prob for all numbers

A = (5,5)

B = (5,1) (5,2)...(5,5) (4,5) (3,5),,,(1,5)

P(A) = 1/25

and P(B) = 9/25

P(AB) = 1/25 not equal to P(A) *P(B)

Hence A and B are not independent

C = atleast one in 1

= (1,1)...(1,5) (2,1) (3,1) (4,1) (5,1)

P(C) = 9/25

P(AC) = 0 not equal to P(A) * P(C) hence A, C are not independent

b) D = (2,5) (3,4) (4,3) (5,2)

E = (1,2) (2,1) (3,2) (2,3) (4,3) (3,4) (5,4) (4,5)

F = (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5)

P(EFD) = 0

BUt P(E) = 8/25 and P(F)=2/5

Hence E and F are not conditionally independent given D

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