Question

a number cube with faces labeled 1 to 6 is rolled once.
the number rolled will be recorded as the outcome.
consider the following events.
event a: the number rolled is odd.
event b: the number rolled is less than 4.
give the outcomes for each of the following events.
if there is more than one element in the set, separate them with commas.
(a) event \a and b\:
(b) event \a or b\:
(c) the complement of the event a:

Explanation:

Step1: Determine the sample space

The sample space \( S=\{1,2,3,4,5,6\} \)

Step2: Find the elements of event \( A \) and event \( B \)

Event \( A=\{1,3,5\} \) (odd - numbered outcomes). Event \( B = \{1,2,3\} \) (outcomes less than \( 4 \))

Part (a): Event “\( A \) and \( B \)”

“\( A \) and \( B \)” is the intersection \( A\cap B \).
\( A\cap B=\{x|x\in A\text{ and }x\in B\} \)
Since \( A = \{1,3,5\} \) and \( B=\{1,2,3\} \), \( A\cap B=\{1,3\} \)

Part (b): Event “\( A \) or \( B \)”

“\( A \) or \( B \)” is the union \( A\cup B \).
\( A\cup B=\{x|x\in A\text{ or }x\in B\} \)
\( A\cup B=\{1,2,3,5\} \)

Part (c): Complement of event \( A \)

The complement of \( A \), denoted as \( A^c\) or \(\overline{A}\), is \( A^c=\{x|x\in S\text{ and }x
otin A\} \)
Since \( S = \{1,2,3,4,5,6\} \) and \( A=\{1,3,5\} \), \( A^c=\{2,4,6\} \)

Answer:

(a) \( \{1,3\} \)
(b) \( \{1,2,3,5\} \)
(c) \( \{2,4,6\} \)