To identify the correct symbolic and Venn diagram descriptions for each statement, we can compare the given descriptions to the options provided.
1. At least two of the events A, B, C occur:
The correct symbolic description is E6: (A∩B)∪(A∩C)∪(B∩C).
The correct Venn diagram description is Region 5: A and B overlap, A and C overlap, B and C overlap.
2. At most two of the events A, B, C occur:
The correct symbolic description is E7: (A∩Bc∩Cc)∪(Ac∩B∩Cc)∪(Ac∩Bc∩C).
The correct Venn diagram description is Region 4: A, B, and C have no overlap.
3. None of the events A, B, C occurs:
The correct symbolic description is E2: (A∩B∩C)c.
The correct Venn diagram description is Region 8: The area outside of A, B, and C.
4. All three events A, B, C occur:
The correct symbolic description is E1: A∩B∩C.
The correct Venn diagram description is Region 6: A, B, and C overlap.
5. Exactly one of the events A, B, C occurs:
The correct symbolic description is E4: B∪(Bc∩Cc).
The correct Venn diagram description is the combination of Regions 1, 3, and 7: A only, B only, or C only.
6. Events A and B occur, but C does not occur:
The correct symbolic description is E5: Ac∩Bc∩Cc.
The correct Venn diagram description is the combination of Regions 2 and 5: A and B overlap, C does not occur.
7. Either event B occurs or, if not, then C also does not occur:
The correct symbolic description is E3: A∩B∩Cc.
The correct Venn diagram description is the combination of Regions 1 and 8: A and B overlap, C does not occur.
Therefore, the correct symbolic and Venn diagram descriptions for each statement are as follows:
1. At least two of the events A, B, C occur:
Symbolic description: E6: (A∩B)∪(A∩C)∪(B∩C)
Venn diagram description: Region 5
2. At most two of the events A, B, C occur:
Symbolic description: E7: (A∩Bc∩Cc)∪(Ac∩B∩Cc)∪(Ac∩Bc∩C)
Venn diagram description: Region 4
3. None of the events A, B, C occurs:
Symbolic description: E2: (A∩B∩C)c
Venn diagram description: Region 8
4. All three events A, B, C occur:
Symbolic description: E1: A∩B∩C
Venn diagram description: Region 6
5. Exactly one of the events A, B, C occurs:
Symbolic description: E4: B∪(Bc∩Cc)
Venn diagram description: Regions 1, 3, and 7
6. Events A and B occur, but C does not occur:
Symbolic description: E5: Ac∩Bc∩Cc
Venn diagram description: Regions 2 and 5
7. Either event B occurs or, if not, then C also does not occur:
Symbolic description: E3: A∩B∩Cc
Venn diagram description: Regions 1 and 8