Question

a standard six - sided die shows a number on each of its sides, so the sample space ( s={1,2,3,4,5,6} ). suppose the die is rolled once. let event ( e ) be the event of rolling the die and it showing an even number on top. let event ( l ) be the event of rolling a number less than 4.
rolling a 3 is an outcome of which of the following events? select all correct answers.
select all that apply:
( square ecup l )
( square e^{c}cap l )
( square ecap l )
( square e^{c}cup l^{c} )

Explanation:

1. First, define the events:

• Sample space \( S=\{1,2,3,4,5,6\} \)
• Event \( E \): rolling an even number, so \( E = \{2,4,6\} \)
• Event \( L \): rolling a number less than 4, so \( L=\{1,2,3\} \)
• Complement of \( E \) (\( E^c \)): \( E^c=\{1,3,5\} \) (numbers that are not even)
• Complement of \( L \) (\( L^c \)): \( L^c = \{4,5,6\} \) (numbers that are not less than 4)

1. Analyze each option:

• Option A: \( E \cup L \) (union of \( E \) and \( L \))
• \( E \cup L=\{1,2,3,4,6\} \) (all elements in \( E \) or \( L \)). Since \( 3 \in L \), \( 3 \in E \cup L \), so rolling a 3 is an outcome of \( E \cup L \).
• Option B: \( E^c \cap L \) (intersection of \( E^c \) and \( L \))
• \( E^c \cap L=\{1,3,5\} \cap \{1,2,3\}=\{1,3\} \). Since \( 3 \in \{1,3\} \), rolling a 3 is an outcome of \( E^c \cap L \).
• Option C: \( E \cap L \) (intersection of \( E \) and \( L \))
• \( E \cap L = \{2,4,6\} \cap \{1,2,3\}=\{2\} \). Since \( 3

otin \{2\} \), rolling a 3 is not an outcome of \( E \cap L \).

• Option D: \( E^c \cup L^c \) (union of \( E^c \) and \( L^c \))
• \( E^c \cup L^c=\{1,3,5\} \cup \{4,5,6\}=\{1,3,4,5,6\} \). Since \( 3 \in \{1,3,4,5,6\} \), rolling a 3 is an outcome of \( E^c \cup L^c \).

Answer:

A. \( E \cup L \)
B. \( E^c \cap L \)
D. \( E^c \cup L^c \)