Question

1. A random experiment consists of throwing a pair of dice, say a red die and a green die, simultaneously. They are standard 6-sided dice with one to six dots on different faces. Describe the sample space.

2. For the same experiment, let E be the event that the sum of the numbers of spots on the two dice is an odd number. Write E as a subset of the sample space, i.e., list the outcomes in E.

3. List the outcomes in the event F = ”the sum of the spots is a multiple of 3”.

4. Find ∼F, E ∪ F, EF = E ∩ F, and E∼F.

5. Assume that the outcomes of this experiment are equally likely. Find the probability of each of the events in # 4.

Answer 1

1)

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)]

n(S) = 36

2)

E = [(1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5)]

n(E) = 18

3)

F = [(1,2),(2,1),(2,4), (4,2),(3,3),(1,5),(5,1),(3,6),(6,3),(5,4),(4,5),(6,6)]

n(F) = 12

4)

~F = [(1, 1), (1, 3), (1, 4), (1, 6), (2, 2), (2, 3), (2, 5), (2, 6) ,(3, 1), (3, 2), (3, 4), (3, 5), (4, 1),   (4, 3), (4, 4), (4, 6),    (5, 2), (5, 3), (5, 5), (5, 6),(6, 1), (6, 2), (6, 4), (6, 5)]

n(F^c ) = 24

EUF = [(1,2), (1,4), (2,1), (2,3),(2,4), (2,5), (3,2), (3,4), (3,6), (4,1), (4,2),(3,3),(1,5),(5,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5) ,(6,6)]

n(E U F) = 24

EF=[(1,2), (2,1),(3,6),(4,5), (5,4) , (6,3)]

n(EF) = 6

E~F= [(1, 4),(1, 6), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1),   (4, 3), (5, 2), (5, 6),(6, 1), (6, 5)]

n(E~F) = 12

5.