Using the given information, construct the corresponding Venn Diagram to include the number of households surveyed in each mutually exclusive region. Total households = 1250, Aggie... Using the given information, construct the corresponding Venn Diagram to include the number of households surveyed in each mutually exclusive region. Total households = 1250, Aggie fans = 950, Longhorn fans = 275, Neither = 100.
Understand the Problem
The problem describes a survey of households in Texas regarding their allegiance to the Aggies (A) and Longhorns (L). Given the total number of households surveyed, the number of households with Aggie fans, the number with Longhorn fans, and the number with neither, we need to construct a Venn diagram and determine the number of households in each mutually exclusive region of the diagram.
Answer
- Aggie only: 875 - Longhorn only: 200 - Both Aggie and Longhorn: 75 - Neither Aggie nor Longhorn: 100
Answer for screen readers
- Aggie only: 875
- Longhorn only: 200
- Both Aggie and Longhorn: 75
- Neither Aggie nor Longhorn: 100
Steps to Solve
- Calculate the number of households that have at least one fan
Since 100 households have neither Aggie nor Longhorn fans, subtract this number from the total number of households surveyed to find the number of households that have at least one type of fan.
$1250 - 100 = 1150$
- Use the principle of inclusion-exclusion
Let $n(A)$ be the number of households with Aggie fans, $n(L)$ be the number of households with Longhorn fans, and $n(A \cup L)$ be the number of households with either Aggie or Longhorn fans or both. Then,
$n(A \cup L) = n(A) + n(L) - n(A \cap L)$
We know that: $n(A \cup L) = 1150$ $n(A) = 950$ $n(L) = 275$
We want to find $n(A \cap L)$, which represents the number of households with both Aggie and Longhorn fans.
- Solve for the intersection
Substitute the known values into the inclusion-exclusion formula and solve for $n(A \cap L)$:
$1150 = 950 + 275 - n(A \cap L)$ $1150 = 1225 - n(A \cap L)$ $n(A \cap L) = 1225 - 1150$ $n(A \cap L) = 75$
- Calculate the number of households with only Aggie fans
Subtract $n(A \cap L)$ from $n(A)$ to find the number of households with only Aggie fans:
$n(A \text{ only}) = n(A) - n(A \cap L)$ $n(A \text{ only}) = 950 - 75 = 875$
- Calculate the number of households with only Longhorn fans
Subtract $n(A \cap L)$ from $n(L)$ to find the number of households with only Longhorn fans:
$n(L \text{ only}) = n(L) - n(A \cap L)$ $n(L \text{ only}) = 275 - 75 = 200$
- Populate the Venn Diagram
- The region representing households with only Aggie fans (A only) has 875 households.
- The region representing households with only Longhorn fans (L only) has 200 households.
- The region representing households with both Aggie and Longhorn fans ($A \cap L$) has 75 households.
- The region outside both circles (neither A nor L) has 100 households.
- Aggie only: 875
- Longhorn only: 200
- Both Aggie and Longhorn: 75
- Neither Aggie nor Longhorn: 100
More Information
The principle of inclusion-exclusion is a fundamental counting technique used in set theory to determine the number of elements in the union of multiple sets. In this specific problem, it helps us accurately determine overlap between Aggie and Longhorn fans.
Tips
A common mistake is not subtracting the intersection when calculating the number of households with only Aggie or only Longhorn fans. For example, if you just put 950 for Aggie only, you are including those who are also Longhorn fans. Another mistake is not accounting for the households that have neither fan when calculating the total number of households with at least one type of fan.
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