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 a. are events a and b mutually exclusive? a. yes. these events do not share any common outcomes. b. yes. at least the outcome where the die comes up 6 is common for these events. c. no. these events do not share any common outcomes. d. no. at least the outcome where the die comes up 6 is common for these events. 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.

Explanation:

Step1: Recall the definition of mutually - exclusive events

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

Step2: Analyze events A and B

Event A: The die comes up even, so $A=\{2,4,6\}$. Event B: The die comes up 3 or more, so $B = \{3,4,5,6\}$. The common outcomes of A and B are 4 and 6. Since there are common outcomes, A and B are not mutually exclusive.

Step3: Analyze events B and C

Event B: $B=\{3,4,5,6\}$. Event C: The die comes up at most 3, so $C=\{1,2,3\}$. The common outcome of B and C is 3. Since there is a common outcome, B and C are not mutually exclusive.

Answer:

a. D. No. At least the outcome where the die comes up 6 is common for these events.
b. B. No. At least the outcome where the die comes up 3 is common for these events.