Topic
A team of young volunteers consists of 27 people from the following provinces (cities): Kon Tum, Gia Lai, Dak Lak, Dak Nong, Lam Dong, Phu Yen, Khanh Hoa, Ninh Thuan, Binh Thuan, Ba Ria – Vung Tau , Binh Duong, Binh Phuoc, Dong Nai, Tay Ninh, Long An, Tien Giang, Vinh Long, Ben Tre, Dong Thap, Tra Vinh, An Giang, Can Tho, Hau Giang, Bac Lieu, Soc Trang, Kien Giang and Ca Quickly; Each province has exactly one team member.
Randomly select 3 members of the team to assign tasks first. Calculate the probability of each of the following events:
a) A: “Three members were selected from the Central Highlands”
b) B: “Three members were selected from the South Central Coast”
c) C: “Three members were selected from the Southeast”
Solution method – See details
The probability of event A being a number, symbol \(P\left( A \right)\) is determined by the formula: \(P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( \Omega \right)}}\), where \(n\left( A \right)\) and \(n\left( \Omega \right)\) denote the number of elements of set A and \(\Omega \) respectively.
Detailed explanation
Choose 3 provinces out of 27 cities \( \Rightarrow n\left( \Omega \right) = C_{27}^3\)
a) A: “Three members are selected from the Central Highlands”: there are 5 provinces: Kon Tum, Gia Lai, Dak Lak, Dak Nong, Lam Dong \( \Rightarrow n\left( A \right) = C_5^3 = 10\)
\( \Rightarrow P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( \Omega \right)}} = \frac{{10}}{{ C_{27}^3}} = \frac{2}{{585}}\)
b) B: “Three selected members are from the South Central Coast”: there are 4 provinces: Phu Yen, Khanh Hoa, Ninh Thuan, Binh Thuan \( \Rightarrow n\left( B \right) = C_4^3 = 4\)
\( \Rightarrow P\left( B \right) = \frac{{n\left( B \right)}}{{n\left( \Omega \right)}} = \frac{4}{{C_{ 27}^3}} = \frac{4}{{2925}}\)
c) C: “Three selected members are from the Southeast”: there are 5 provinces: Ba Ria – Vung Tau, Binh Duong, Binh Phuoc, Dong Nai, Tay Ninh \( \Rightarrow n\left( C \right) = C_5^3 = 10\)
\( \Rightarrow P\left( C \right) = \frac{{n\left( C \right)}}{{n\left( \Omega \right)}} = \frac{{10}}{{ C_{27}^3}} = \frac{2}{{585}}\)
d) D: “Three selected members are from the Mekong Delta”: there are 13 provinces: Long An, Tien Giang, Vinh Long, Ben Tre, Dong Thap, Tra Vinh, An Giang, Can Tho, Hau Giang, Bac Lieu, Soc Trang, Kien Giang, Ca Mau \( \Rightarrow n\left( D \right) = C_{13}^3 = 286\)
\( \Rightarrow P\left( D \right) = \frac{{n\left( D \right)}}{{n\left( \Omega \right)}} = \frac{{286}}{{ C_{27}^3}} = \frac{{22}}{{225}}\)
