Soru

Zorluk: OrtaLogical Venn Diagrams

In a department of 150150 administrative officers, a survey was conducted regarding specialized training in three skills: Cyber Security (CC), Data Analytics (DD), and Disaster Management (MM). The survey revealed the following data:
- 7575 officers are trained in Cyber Security
- 6565 officers are trained in Data Analytics
- 5555 officers are trained in Disaster Management
- 3030 officers are trained in both Cyber Security and Data Analytics
- 2525 officers are trained in both Data Analytics and Disaster Management
- 2020 officers are trained in both Cyber Security and Disaster Management
- 1010 officers are trained in all three areas

Based on this information, which of the following statements are correct?

  1. The number of officers trained only in Cyber Security is 3535.Cevap
  2. B
    The number of officers trained in exactly two of these subjects is 5555.
  3. The number of officers who are not trained in any of these three subjects is 2020.Cevap
  4. D
    The number of officers trained in at least two subjects is 6565.

Cevap

The statements confirming that 3535 officers are trained only in Cyber Security and that 2020 officers are not trained in any of the three subjects are correct.
The statement specifying that 3535 officers are trained only in Cyber Security is correct because 75(20+10+10)=3575 - (20 + 10 + 10) = 35. The statement specifying that 2020 officers are not trained in any of the three subjects is also correct because the total union of officers with at least one training is 130130, leaving 150130=20150 - 130 = 20.

Adım Adım Çözüm

1
Calculate the number of officers trained in exactly two areas and all three areas
Only Cyber Security and Data Analytics = 3010=2030 - 10 = 20; Only Data Analytics and Disaster Management = 2510=1525 - 10 = 15; Only Cyber Security and Disaster Management = 2010=1020 - 10 = 10; All three = 1010.
To avoid double-counting the triple-intersection region (1010), it must be subtracted from each pairwise intersection.
2
Calculate officers trained in only one area
Only Cyber Security = 75(20+10+10)=3575 - (20 + 10 + 10) = 35; Only Data Analytics = 65(20+15+10)=2065 - (20 + 15 + 10) = 20; Only Disaster Management = 55(10+15+10)=2055 - (10 + 15 + 10) = 20.
Subtract all overlapping regions from the total count of each individual set.
3
Find the total number of officers trained in at least one subject (Union of 3 sets)
Total trained = 35+20+20+20+15+10+10=13035 + 20 + 20 + 20 + 15 + 10 + 10 = 130.
Sum up all seven mutually exclusive regions.
4
Determine the number of officers not trained in any of the three subjects
None = 150130=20150 - 130 = 20.
Subtract the total in the union from the total population of officers (150150).

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets
Bu soruyu puanla