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 b. are events b and c mutually exclusive? a. yes. these events do not share any common outcomes. b. no. at least the outcome where the die comes up 3 is common for these events. c. no. these events do not share any common outcomes. d. yes. at least the outcome where the die comes up 3 is common for these 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.

Explanation:

Step1: Recall definition of mutually - exclusive events

Two or more events are mutually exclusive if they cannot occur at the same time, i.e., they have no common outcomes.

Step2: Analyze events B and C

Event B = {3, 4, 5, 6}, Event C = {1, 2, 3}. The outcome 3 is common to both B and C, so they are not mutually exclusive.

Step3: Analyze events A, C, and D

Event A = {2, 4, 6}, Event C = {1, 2, 3}, Event D = {6}. The outcome 2 is common to A and C, so A, C, and D are not mutually exclusive.

Step4: Analyze if there are three mutually - exclusive events among A, B, C, and D

A = {2, 4, 6}, B = {3, 4, 5, 6}, C = {1, 2, 3}, D = {6}. A and B share 4 and 6, C and D share 6. So, there are no three mutually - exclusive events among them.

Answer:

b. B. No. At least the outcome where the die comes up 3 is common for these events.
c. A. No. At least the outcome where the die comes up 2 is common for A and C.
d. A. No. A and B are not mutually exclusive and C and D are not mutually exclusive.