Question

In a soccer survey
90 people supported kaizer chiefs
80 supported Orlando pirates
5 people supported both teams
10 people who did not support either teams


A. Draw a venn diagram to illustrate the above information.

B.How many people supported:
1. Kaizer chiefs in total
2. Orlando pirates total

C. How many people were there in their survey ?

D. Determine whether the events involved are inclusive or mutually exclusive

E. Determine the probability that a person selected at random will support Kaizer chiefs only

F. Determine the probability that a person selected at random will support Orlando pirates only.


G. Determine the probability that a person selected at random will support both teams

Answer 1

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:

1. Let circle \( A \) represent the supporters of Kaizer Chiefs.
2. Let circle \( B \) represent the supporters of Orlando Pirates.
3. 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:
1. **Kaizer Chiefs in total**:
- Total supporters = Supporters only of Kaizer Chiefs + Supporters of both teams
- Total for Kaizer Chiefs = \( 85 + 5 = 90 \)

2. **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:
1. Total people supported Kaizer Chiefs = 90
2. Total people supported Orlando Pirates = 80
3. Total people surveyed = 175
4. Events are inclusive.
5. Probability support Kaizer Chiefs only ≈ 0.486
6. Probability support Orlando Pirates only ≈ 0.429
7. Probability support both teams ≈ 0.029

If you need any further explanation or additional calculations, feel free to ask!