To tackle the survey analysis based on the provided information, we'll break it down step-by-step.
A. Venn Diagram
To illustrate the problem using a Venn diagram:
- Let circle \( A \) represent the supporters of Kaizer Chiefs.
- Let circle \( B \) represent the supporters of Orlando Pirates.
- The intersection of the circles \( A \cap B \) represents the supporters of both teams.
From the data:
- Supporters of Kaizer Chiefs (A) = 90
- Supporters of Orlando Pirates (B) = 80
- Supporters of both teams (A ∩ B) = 5
- Supporters of neither = 10
The counts for the Venn diagram are:
- Only Kaizer Chiefs (A only) = \( 90 - 5 = 85 \)
- Only Orlando Pirates (B only) = \( 80 - 5 = 75 \)
So, we can summarize it as follows:
- Supporters only of Kaizer Chiefs = 85
- Supporters only of Orlando Pirates = 75
- Supporters of both teams = 5
Make sure to adjust the diagram accordingly.
B. How many people supported:
-
Kaizer Chiefs in total:
- Total supporters = Supporters only of Kaizer Chiefs + Supporters of both teams
- Total for Kaizer Chiefs = \( 85 + 5 = 90 \)
-
Orlando Pirates total:
- Total supporters = Supporters only of Orlando Pirates + Supporters of both teams
- Total for Orlando Pirates = \( 75 + 5 = 80 \)
C. How many people were there in their survey?
Total people surveyed:
- People who support Kaizer Chiefs + People who support Orlando Pirates + People who support both + People who support neither
So the total is:
- \( 85 + 75 + 5 + 10 = 175 \)
D. Determine whether the events involved are inclusive or mutually exclusive
- The events are inclusive because some individuals support both teams (the intersection is non-empty). If they were mutually exclusive, it would mean that no individual could support both teams at the same time.
E. Determine the probability that a person selected at random will support Kaizer Chiefs only
Probability (support Kaizer Chiefs only) = \[ \frac{\text{Supporters only of Kaizer Chiefs}}{\text{Total surveyed}} = \frac{85}{175} \] Calculating: \[ \frac{85}{175} = 0.4857 , \text{(approximately)} \]
F. Determine the probability that a person selected at random will support Orlando Pirates only
Probability (support Orlando Pirates only) = \[ \frac{\text{Supporters only of Orlando Pirates}}{\text{Total surveyed}} = \frac{75}{175} \] Calculating: \[ \frac{75}{175} = 0.4286 , \text{(approximately)} \]
G. Determine the probability that a person selected at random will support both teams
Probability (support both teams) = \[ \frac{\text{Supporters of both teams}}{\text{Total surveyed}} = \frac{5}{175} \] Calculating: \[ \frac{5}{175} = 0.0286 , \text{(approximately)} \]
Summary of Results:
- Total people supported Kaizer Chiefs = 90
- Total people supported Orlando Pirates = 80
- Total people surveyed = 175
- Events are inclusive.
- Probability support Kaizer Chiefs only ≈ 0.486
- Probability support Orlando Pirates only ≈ 0.429
- Probability support both teams ≈ 0.029
If you need any further explanation or additional calculations, feel free to ask!