In this problem, you are given descriptions in words of certain events (e.g., "at least one of the events A,B,C occurs"). For each one of these descriptions, identify the correct symbolic description in terms of A,B,C from Events E1-E7 below. Also identify the correct description in terms of regions (i.e., subsets of the sample space Ω) as depicted in the Venn diagram below. (For example, Region 1 is the part of A outside of B and C.)

Symbolic descriptions:

Event E1: A∩B∩C

Event E2: (A∩B∩C)c

Event E3: A∩B∩Cc

Event E4: B∪(Bc∩Cc)

Event E5: Ac∩Bc∩Cc

Event E6: (A∩B)∪(A∩C)∪(B∩C)

Event E7: (A∩Bc∩Cc)∪(Ac∩B∩Cc)∪(Ac∩Bc∩C)

At least two of the events A, B, C occur.

Status: incorrect

Status: incorrect
At most two of the events A, B, C occur.

Status: incorrect

Status: incorrect
None of the events A, B, C occurs.

Status: incorrect

Status: incorrect
All three events A, B, C occur.

Status: incorrect

Status: correct
Exactly one of the events A, B, C occurs.

Status: incorrect

Status: incorrect
Events A and B occur, but C does not occur.

Status: incorrect

Status: correct
Either event B occurs or, if not, then C also does not occur.

Status: incorrect

Status: incorrect

At least two of the events A, B, C occur.

Event E6
Regions: 2 4 5 6

At most two of the events A, B, C occur.
Event E2
Regions: 1 2 3 5 6 7 8

None of the events A, B, C occurs.
Event E5
Region: 8

All three events A, B, C occur.
Event E1
Region: 4

Exactly one of the events A, B, C occurs.
Event E7
Regions: 1 3 7

Events A and B occur, but C does not occur.
Event E3
Region: 2

Either event B occurs or, if not, then C also does not occur.
Event E4
Regions: 1 2 3 4 6 8

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

To identify the correct symbolic description and description in terms of regions for each given event, we can analyze the statements one by one.

1. At least two of the events A, B, C occur.

The symbolic description for this statement would be E6: (A∩B)∪(A∩C)∪(B∩C). This is because E6 represents the union of all possible combinations of two events occurring.

The description in terms of regions for this statement would be Region 6 in the Venn diagram, which is the union of regions where at least two events occur.

2. At most two of the events A, B, C occur.

The symbolic description for this statement would be E7: (A∩Bc∩Cc)∪(Ac∩B∩Cc)∪(Ac∩Bc∩C). This is because E7 represents the union of all possible combinations where at most two events occur.

The description in terms of regions for this statement would be the union of regions 1, 2, 3, 4, 5, and 7 in the Venn diagram, which represent the combinations where at most two events occur.

3. None of the events A, B, C occurs.

The symbolic description for this statement would be (A∩B∩C)c, which is the complement of E1. This means that none of the events A, B, C occur.

The description in terms of regions for this statement would be the region outside the circles A, B, and C in the Venn diagram, which represents the possibility of none of the events occurring.

4. All three events A, B, C occur.

The symbolic description for this statement would be E1: A∩B∩C. This means that all three events occur simultaneously.

The description in terms of regions for this statement would be the region where all three circles A, B, and C intersect in the Venn diagram, which represents the possibility of all three events occurring.

5. Exactly one of the events A, B, C occurs.

The symbolic description for this statement would be E4: B∪(Bc∩Cc). This means that exactly one event occurs, which can be B or the intersection of the complements of B and C.

The description in terms of regions for this statement would be the union of regions 2, 3, and 4 in the Venn diagram, which represent the combinations where exactly one event occurs.

6. Events A and B occur, but C does not occur.

The symbolic description for this statement would be E3: A∩B∩Cc. This means that A and B occur, but C does not occur.

The description in terms of regions for this statement would be the intersection of regions 2 and 4 in the Venn diagram, which represent the combinations where A and B occur, but C does not occur.

7. Either event B occurs or, if not, then C also does not occur.

The symbolic description for this statement would be E5: Ac∩Bc∩Cc. This means that either B occurs or, if not, both B and C do not occur.

The description in terms of regions for this statement would be the intersection of regions 3 and 5 in the Venn diagram, which represent the combinations where either B occurs or both B and C do not occur.

By matching the given statements with the appropriate symbolic descriptions and descriptions in terms of regions, we can determine the correct answers.