Question

when one die is rolled, six outcomes are possible. these outcomes are those where the die shows each value between 1 and 6. complete parts (a) through (d). a = event the die comes up even b = event the die comes up 3 or more c = event the die comes up at most 3 d = event the die comes up 6 d. yes. at least the outcome where the die comes up 3 is common for those events. c. are events a, c, and d mutually exclusive? a. no. at least the outcome where the die comes up 2 is common for a and c. b. no. these events do not share any common outcomes. c. yes. these events do not share any common outcomes. d. yes. at least the outcome where the die comes up 2 is common for a and c. d. are there three mutually exclusive events among a, b, c, and d? a. no. a and b are not mutually exclusive and c and d are not mutually exclusive. b. no. a and c are not mutually exclusive and b and d are not mutually exclusive. c. yes. a, c, and d are mutually exclusive. d. yes. a, b, and c are mutually exclusive. are all four of the events mutually exclusive? a. no. events a and c are not mutually exclusive because they share at least one common outcome. b. no. events c and d are not mutually exclusive because they do not share any common outcomes. c. no. events a and d are not mutually exclusive because they do not share any common outcomes. d. yes. all four events are mutually exclusive because they share at least one common outcome.

Explanation:

Step1: Define the events

A = {2, 4, 6}, B = {3, 4, 5, 6}, C = {1, 2, 3}, D = {6}

Step2: Analyze part c

For events A, C and D to be mutually - exclusive, they should have no common outcomes. But A = {2, 4, 6} and C = {1, 2, 3} have 2 as a common outcome. So they are not mutually exclusive.

Step3: Analyze part d

Check pairs of events:

• A and B: A = {2, 4, 6}, B = {3, 4, 5, 6}, they have 4 and 6 in common.
• A and C: A = {2, 4, 6}, C = {1, 2, 3}, they have 2 in common.
• B and C: B = {3, 4, 5, 6}, C = {1, 2, 3}, they have 3 in common.
• B and D: B = {3, 4, 5, 6}, D = {6}, they have 6 in common.
• C and D: C = {1, 2, 3}, D = {6}, no common outcomes.

Since A and C are not mutually exclusive and B and D are not mutually exclusive, there are no three mutually - exclusive events among A, B, C and D.

Step4: Analyze the last part

Since A and C have a common outcome (2), all four events A, B, C and D are not mutually exclusive.

Answer:

c. A. No. At least the outcome where the die comes up 2 is common for A and C.
d. B. No. A and C are not mutually exclusive and B and D are not mutually exclusive.
Last part: A. No. Events A and C are not mutually exclusive because they share at least one common outcome.