Question

you roll a die with the sample space s = {1, 2, 3, 4, 5, 6}. you define a as {1, 3, 5}, b as {2, 3, 4, 5, 6}, c as {4, 5}, and d as {3, 5, 6}. determine which of the following events are exhaustive and/or mutually exclusive.
a. a and b
b. a and c
c. a and d
d. b and c
exhaustive
mutually exclusive

Explanation:

Step1: Recall definitions

Exhaustive events: Their union is the sample - space. Mutually exclusive events: They have no common elements.

Step2: Analyze A and B

$A=\{1,3,5\}$, $B = \{2,3,4,5,6\}$. $A\cup B=\{1,2,3,4,5,6\}=S$, but $A\cap B = \{3,5\}
eq\varnothing$, so not mutually exclusive.

Step3: Analyze A and C

$A=\{1,3,5\}$, $C=\{4,5\}$. $A\cup C=\{1,3,4,5\}
eq S$, and $A\cap C = \{5\}
eq\varnothing$, so neither exhaustive nor mutually exclusive.

Step4: Analyze A and D

$A=\{1,3,5\}$, $D=\{3,5,6\}$. $A\cup D=\{1,3,5,6\}
eq S$, and $A\cap D=\{3,5\}
eq\varnothing$, so neither exhaustive nor mutually exclusive.

Step5: Analyze B and C

$B = \{2,3,4,5,6\}$, $C=\{4,5\}$. $B\cup C=\{2,3,4,5,6\}
eq S$, and $B\cap C=\{4,5\}
eq\varnothing$, so neither exhaustive nor mutually exclusive.

Answer:

None of the given pairs are both exhaustive and mutually exclusive.