Question

1) Consider rolling a die two times.

a.What is the sample space ?2 if the outcome of interest is the maximum of the two numbers observed?
B.What is the sample space ?3 if the outcome of interest is the difference between the first and the second numbers? (Hint: ?1 − ?2)
C.What is the sample space ?4 if the outcome of interest is the distance between the two number?
D.Let ? be the event in which the distance between the two observed numbers is 1. Let ?? be the expression of ? (the list of its members) based on the ??h sample space (??). Express ? in as many ways as possible. (Find ?1,?2,?3 and ?4 if possible).
E.Let ? be the event in which the second roll is greater than the first roll. Express ? in as many ways as possible (?1, ... , ?4)
F.Find ? ∪ ?, ? ∩ ? and ?\?. Use subscript 1 to 2 to identify which sample space you are using.

g. What is the sample space S if the outcome of interest is the pair of numbers observed?

Answer 1

$1.A.~S_2=\{1,2,3,4,5,6\}\\ B.~S_{3}=\{-5,-4,-3,-2,-1,0,1,2,3,4,5\}\\ C.~S_4=\{0,1,2,3,4,5\}\\ D.~A=\{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(4,5),(5,4),(5,6),(6,5)\};\\ (x,y):~Outcome~when~a~die~is~rolled~twice~where~x=outcome~from~first~roll,\\ y=outcome~from~2nd~roll.\\ E.~B=\{(1,2),(1,3),(2,3),(1,4),(2,4),(3,4),(1,5),(2,5),(3,5),\\ (4,5),(1,6),(2,6),(3,6),(4,6),(5,6)\}\\ F.~A\cup B=\{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(4,5),(5,4),(5,6),(6,5),(1,3),\\ (1,4),(2,4),(1,5),(2,5),(3,5),(1,6),(2,6),(3,6),(4,6)\}\\ A\cap B=\{(1,2),(2,3),(3,4),(4,5),(5,6)\}\\ A-B=\{(2,1),(3,2),(4,3),(5,4),(6,5)\}\\ g.~S=\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),\\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}$