A sample of 100 Young Christian Union voters revealed the following concerning three candidates; Ali, Bungei and Chiru, who were running for the Y.C.S Party Chairman, Secretary and Treasurer respectively. 14 preferred both Ali and Bungei 49 preferred Ali or Bungei but not Chiru 21 preferred Bungei but not Chiru or Ali 61 preferred Bungei or Chiru but not Ali 32 preferred Chiru but not Ali or Bungei 7 preferred Ali and Chiru but not Bungei.
i) Present the information in a Venn diagram.
ii) Determine the number of voters that were in favor of all the three candidates.
iii) Assume that every member of Y.C.S voted for at least one candidate. Determine the candidate that went unopposed if a rule of 50% majority were used in such a decision.
Kwani ni kesho
i) Here is the information presented in a Venn diagram:
```
A B C
(Ali) (Bungei) (Chiru)
| | |
| | |
__|____ ____|____ ____|____
| | | | | |
| 7 | | 7 | | 7 |
| | | | | |
|____|__| |___|___| |___|___|
| | |
(14) (21) (32) (49) (61)
```
ii) To determine the number of voters that were in favor of all three candidates (Ali, Bungei, and Chiru), we need to look at the overlapping regions of the Venn diagram. From the diagram, we can see that the number of voters in favor of all three candidates is 7.
iii) If the rule of 50% majority were used, we would need to find the candidate with the highest number of overall preferences. From the Venn diagram, we can see that Bungei has the highest number of preferences, with 21 + 7 + 49 = 77 voters. Therefore, Bungei would be the candidate who went unopposed if the rule of 50% majority were used.
i) The information can be presented in a Venn diagram as follows:
A B
(Ali) (Bungei)
| |
| |
| |
| |
AB B
(Ali & Bungei) (Bungei only)
| |
| |
| |
| |
AC C
(Ali & Chiru) (Chiru only)
| |
| |
| |
| |
ABC
(All three)
ii) To determine the number of voters that were in favor of all three candidates, we can look at the overlapping region, which is represented by ABC. Based on the information given, we do not have a specific number for this overlapping region. It is indicated that 7 preferred Ali and Chiru but not Bungei, but we cannot determine if these 7 individuals are part of the ABC region or not. Therefore, we cannot determine the exact number of voters who preferred all three candidates.
iii) If a rule of 50% majority were used to determine the candidate that went unopposed, we need to calculate the number of voters who supported each candidate.
- To calculate the number of voters who supported Ali, we sum the individuals who preferred Ali and the individuals who preferred both Ali and Bungei, excluding any overlap with Chiru:
Ali = 49 (preferred Ali or Bungei but not Chiru) + 7 (preferred Ali and Chiru but not Bungei) = 56
- To calculate the number of voters who supported Bungei, we sum the individuals who preferred Bungei and the individuals who preferred both Ali and Bungei, excluding any overlap with Chiru:
Bungei = 14 (preferred both Ali and Bungei) + 21 (preferred Bungei but not Chiru or Ali) + 61 (preferred Bungei or Chiru but not Ali) = 96
- To calculate the number of voters who supported Chiru, we sum the individuals who preferred Chiru and the individuals who preferred Bungei or Chiru, excluding any overlap with Ali:
Chiru = 32 (preferred Chiru but not Ali or Bungei) + 61 (preferred Bungei or Chiru but not Ali) + 7 (preferred Ali and Chiru but not Bungei) = 100
Based on these calculations, we can see that Bungei had the highest number of supporters (96), which is more than 50% of the total voters (100). Therefore, Bungei would be the candidate that went unopposed.
i) To present the information in a Venn diagram, we need to categorize the preferences of the voters based on the candidates. Let's define three circles to represent the candidates Ali, Bungei, and Chiru.
- Circle A represents Ali.
- Circle B represents Bungei.
- Circle C represents Chiru.
Using the given information, we can determine the following:
- 14 preferred both Ali and Bungei, so this goes in the overlapping region of circles A and B.
- 49 preferred either Ali or Bungei but not Chiru, so this goes in the region of circles A and B but outside circle C.
- 21 preferred Bungei but not Chiru or Ali, so this goes in the region of circle B but outside circles A and C.
- 61 preferred Bungei or Chiru but not Ali, so this goes in the region of circles B and C but outside circle A.
- 32 preferred Chiru but not Ali or Bungei, so this goes in the region of circle C but outside circles A and B.
- 7 preferred Ali and Chiru but not Bungei, so this goes in the overlapping region of circles A and C but outside circle B.
Based on this information, we can draw the Venn diagram as follows:
______________
/ B \
/ ∩ ∩ \
/ \
A C
\ /
\ ∪ ∪ /
\ ______________/
ii) To determine the number of voters who were in favor of all three candidates, we need to find the overlapping region of circles A, B, and C. Looking at the Venn diagram, there is no specific information given for this region. Therefore, we can conclude that there were no voters who preferred all three candidates.
iii) To determine the candidate that went unopposed using a rule of 50% majority, we need to compare the number of voters for each candidate. Let's calculate:
- The number of voters who preferred Ali:
- Preferred both Ali and Bungei: 14
- Preferred either Ali or Bungei but not Chiru: 49
- Preferred Ali and Chiru but not Bungei: 7
Total voters who preferred Ali = 14 + 49 + 7 = 70
- The number of voters who preferred Bungei:
- Preferred both Ali and Bungei: 14
- Preferred either Ali or Bungei but not Chiru: 49
- Preferred Bungei but not Chiru or Ali: 21
- Preferred Bungei or Chiru but not Ali: 61
Total voters who preferred Bungei = 14 + 49 + 21 + 61 = 145
- The number of voters who preferred Chiru:
- Preferred Bungei or Chiru but not Ali: 61
- Preferred Chiru but not Ali or Bungei: 32
- Preferred Ali and Chiru but not Bungei: 7
Total voters who preferred Chiru = 61 + 32 + 7 = 100
Comparing the number of voters for each candidate, we see that Bungei received the highest number of votes (145), followed by Chiru (100), and Ali (70). Since there is no candidate who received more than 50% of the votes, we cannot determine a candidate who went unopposed based on the given information.