Question

A team of five person is formed from 8 boys and 5 girls. The probability that the team contains at least 3 girls is (a) \(\left[(321) /\left({ }^{13} \mathrm{P}_{5}\right)\right]\) (b) \(\left[(321) /\left({ }^{13} \mathrm{C}_{5}\right)\right]\) (c) \(\left[(123) /\left({ }^{13} \mathrm{C}_{5}\right)\right]\) (d) \(\left[(213) /\left({ }^{13} \mathrm{C}_{5}\right)\right]\)

Short answer

The short answer is: The probability that the team contains at least 3 girls is \( \frac{321}{1287} \) which corresponds to option (b).

Step-by-step solution

Calculate the number of ways to make a team with 3, 4, or 5 girls

We can use the combination formula to find the number of ways to choose 3, 4, or 5 girls from a total of 5 girls, and 2, 1, or 0 boys from a total of 8 boys. We'll call these values G3, G4, and G5 respectively. For 3 girls: G3 = \({ }^{5}C_{3} * {}^{8}C_{2}\) For 4 girls: G4 = \({ }^{5}C_{4} * {}^{8}C_{1}\) For 5 girls: G5 = \({ }^{5}C_{5} * {}^{8}C_{0}\)

Calculate the total number of ways to make a team of 5 persons

Now, let's determine the total number of ways to make a team of 5 persons from a group of 13 (8 boys and 5 girls). We'll call this value T. T = \({ }^{13}C_{5}\)

Calculate the probability of having a team with at least 3 girls

Finally, we'll find the probability by dividing the sum of the ways to make a team with at least 3 girls (G3 + G4 + G5) by the total number of ways to make a team of 5 persons (T). Probability = \(\frac{G3 + G4 + G5}{T}\) Plug in the values from step 1 and step 2: Probability = \(\frac{({ }^{5}C_{3} * {}^{8}C_{2}) + ({}^{5}C_{4} * {}^{8}C_{1}) + ({}^{5}C_{5} * {}^{8}C_{0})}{{ }^{13}C_{5}}\) Probability = \(\frac{(10 * 28) + (5 * 8) + (1 * 1)}{1287}\) Probability = \(\frac{321}{1287}\) So, the correct option is: (b) \(\left[(321) /\left({ }^{13} \mathrm{C}_{5}\right)\right]\)

Key concepts

Combinatorics in Team Selection

Combinatorics is the branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as the number of objects in a set or that the order of selection does not matter. In our exercise, we used combinatorics to figure out how many different teams of five could be formed from a group of eight boys and five girls. Specifically, you dealt with the concept of combinations, where the order of the selection is not important.

Using combinatorial reasoning, you calculated the number of possible teams that include at least three girls. You used the combination formula, denoted by \( { }^{n}C_{r} \), which represents the number of ways of choosing \( r \) objects from a total of \( n \) without regard to order. To construct teams that meet the criteria, you combined choices of girls and boys separately and then found the overall number of teams through multiplication.

Permutations and Combinations

To differentiate between permutations and combinations, it's essential to know that permutations consider order while combinations do not. For instance, if you're forming a queue of people, the order in which they stand matters, and that's where permutations come into play. When the order doesn't matter, like selecting a group of people to form a team, we use combinations.

In our exercise, since a team of five persons doesn't change regardless of the order in which we select the members, combinations are used to calculate the number of ways to form the team. Mathematically, the combination for selecting \( r \) items out of \( n \) available is denoted by \( { }^{n}C_{r} = \frac{n!}{r! (n-r)!} \), where \( ! \) stands for factorial, the product of all positive integers up to that number. You applied this to calculate the number of ways to select 3, 4, or 5 girls from 5, and the corresponding number of boys from 8.

Probability Theory

Probability theory is the part of mathematics that studies random events and quantifies the likelihood of different outcomes. The probability of an event is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 denotes a certain event. In the context of our team selection problem, you calculated the probability of forming a team with at least three girls.

To do this, you used the formula for probability, which is \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of equally likely outcomes}} \). 'Favorable outcomes' here means the number of teams with at least 3 girls (which you calculated using combinations), and the 'Total number of equally likely outcomes' is the total number of ways to form any team of 5 from the 13 people. By dividing the count of favorable combinations by the total number of combinations, you accurately found the probability of the team composition, ensuring a clear understanding of the application of both combinatorics and probability theory in solving real-world problems.