A survey was conducted of 1250 households in Texas. 950 households included an Aggie fan (A), while 275 households included a Longhorn fan (L). 100 households did not include any A... A survey was conducted of 1250 households in Texas. 950 households included an Aggie fan (A), while 275 households included a Longhorn fan (L). 100 households did not include any Aggie or Longhorn fans. 1. Using the given information, construct the corresponding Venn Diagram to include the number of households surveyed in each mutually exclusive region. 2. Write the symbolic notation for the event corresponding to the shaded regions below. Then verbally describe the event in the context of the problem.
Understand the Problem
This question involves analyzing survey data using Venn diagrams. The survey asked Texas households about their preference for Aggie (A) and Longhorn (L) sports teams. The first part asks to use the given information to populate a Venn diagram with the number of households in each region. The second part asks for the symbolic notation and a verbal description of the regions in the Venn diagram that are shaded.
Answer
1. Venn Diagram: - Only Aggie fans: 875 - Only Longhorn fans: 200 - Both Aggie and Longhorn fans: 75 - Neither Aggie nor Longhorn fans: 100 2. $L^c$; Households that do not include any Longhorn fans.
Answer for screen readers
- Venn Diagram:
- Only Aggie fans: 875
- Only Longhorn fans: 200
- Both Aggie and Longhorn fans: 75
- Neither Aggie nor Longhorn fans: 100
- Symbolic notation: $L^c$ Verbal description: The event represents households that do not include any Longhorn fans.
Steps to Solve
- Calculate the number of households that are Aggie or Longhorn fans or both
We know the total number of households surveyed is 1250, and 100 households are neither Aggie nor Longhorn fans. Therefore, the number of households that have at least one fan (Aggie or Longhorn) is: $1250 - 100 = 1150$
- Use the principle of inclusion-exclusion to find the intersection
Let $A$ represent the set of households with Aggie fans and $L$ represent the set of households with Longhorn fans. We are given $|A| = 950$ and $|L| = 275$. We also know $|A \cup L| = 1150$ from step 1. Using the principle of inclusion-exclusion: $|A \cup L| = |A| + |L| - |A \cap L|$ Substituting the known values: $1150 = 950 + 275 - |A \cap L|$ $|A \cap L| = 950 + 275 - 1150 = 75$
- Calculate the number of households with only Aggie fans
This is the number of Aggie fans minus the number of households that have both Aggie and Longhorn fans: $|A| - |A \cap L| = 950 - 75 = 875$
- Calculate the number of households with only Longhorn fans
This is the number of Longhorn fans minus the number of households that have both Aggie and Longhorn fans: $|L| - |A \cap L| = 275 - 75 = 200$
- Populate the Venn Diagram
- The number of households with only Aggie fans is 875.
- The number of households with only Longhorn fans is 200.
- The number of households with both Aggie and Longhorn fans is 75.
- The number of households with neither Aggie nor Longhorn fans is 100.
- Determine the symbolic notation for the shaded region
The shaded region includes everything outside of set $L$. This can be represented as the complement of $L$, which is $L^c$.
- Describe the shaded region verbally
The shaded region represents the households that do not have a Longhorn fan.
- Venn Diagram:
- Only Aggie fans: 875
- Only Longhorn fans: 200
- Both Aggie and Longhorn fans: 75
- Neither Aggie nor Longhorn fans: 100
- Symbolic notation: $L^c$ Verbal description: The event represents households that do not include any Longhorn fans.
More Information
Venn diagrams are a useful way to visualize set relationships and solve counting problems. The principle of inclusion-exclusion is fundamental in combinatorics and probability.
Tips
A common mistake is not using the principle of inclusion-exclusion correctly, which leads to errors in determining the number of households in the intersection of the two sets. Another mistake is misinterpreting the word "only" in the problem, and not subtracting the intersection when looking for only Aggie or only Longhorn fans. Finally, misinterpreting the symbolic notation is possible.
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