Correct Option (2)
Let's define the given proportions of villagers:
- Villagers with own houses: 1/2 = 50%
- Villagers cultivating paddy: 1/5 = 20%
- Literate villagers: 1/3 ≈ 33.33%
- Villagers under 25 years of age: 4/5 = 80%
To determine the minimum percentage of overlap between two groups, A and B, we can use the principle of inclusion-exclusion. The minimum overlap is given by P(A) + P(B) - 100%. If this sum is less than 0%, the minimum overlap is 0%.
For Option 2, we consider villagers under 25 years of age (80%) and literate villagers (33.33%).
Minimum overlap = 80% + 33.33% - 100% = 113.33% - 100% = 13.33%.
Since the minimum overlap is 13.33%, it is certain that at least 13.33% of the villagers possess both characteristics (being under 25 and literate). Therefore, the statement "Some villagers under 25 years of age are literate" is certainly correct.
Incorrect Options:
- Option 1: All the villagers who have their own houses are literate.
- Villagers with own houses constitute 50% of the total, while literate villagers constitute approximately 33.33%.
- For all villagers with houses (50%) to be literate, the literate population would need to be at least 50%. However, it is only 33.33%. This makes the statement impossible and thus certainly incorrect. The maximum possible overlap between these two groups is 33.33%.
- Option 3: Only half of the villagers who cultivate paddy are literate.
- Villagers cultivating paddy are 20%, and literate villagers are 33.33%.
- "Only half of the villagers who cultivate paddy" implies that exactly 10% (half of 20%) of the total villagers are both paddy cultivators and literate.
- The actual overlap between these two groups can range from a minimum of 0% (since 20% + 33.33% - 100% is negative) to a maximum of 20% (the smaller of the two percentages).
- Since the overlap is not fixed at 10%, this statement is not certainly correct.
- Option 4: No villager under 25 years of age has his own house.
- Villagers under 25 years of age are 80%, and villagers with own houses are 50%.
- To find the minimum overlap: 80% + 50% - 100% = 130% - 100% = 30%.
- This indicates that at least 30% of the villagers are both under 25 and have their own house. Therefore, the statement "No villager under 25 years of age has his own house" is certainly incorrect.