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Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It plays a crucial role in probability theory as it helps us count the number of possible outcomes in a systematic way.
When dealing with our problem, we use combinatorics to determine how many ways we can choose subsets of individuals from a larger group. One tool used in combinatorics is the binomial coefficient, represented by \( \binom{n}{k} \), which tells us how many ways we can choose \( k \) items from \( n \) items without considering the order of selection.
In our exercise, we need to calculate the total ways to select 3 people from 10, which is \( \binom{10}{3} = 120 \). Then, for instance, to find out how many ways 2 people can prefer football out of 7, and 1 person can prefer basketball out of 3, we calculate \( \binom{7}{2} = 21 \) and \( \binom{3}{1} = 3 \) respectively. These calculations are key to solving the probability of exactly 2 people preferring football.

Statistical methods provide us with techniques to collect, analyze, interpret, and present data. In probability theory, these methods help determine the likelihood of certain events happening.
To solve the problem at hand, we applied statistical methods by using the calculated combinations to determine probabilities. The probability of an event is found by dividing the number of favorable outcomes by the total possible outcomes. In our exercise, this means taking the combinations from our combinatorics calculations and using them as part of the probability formula.

• For exactly 2 individuals preferring football, we calculate the probability as \( P = \frac{\binom{7}{2} \times \binom{3}{1}}{\binom{10}{3}} = \frac{63}{120} = \frac{21}{40} \).
• For the majority preferring football, we sum the probabilities of 2 and 3 individuals preferring football: \( P = \frac{63 + 35}{120} = \frac{98}{120} = \frac{49}{60} \).

The understanding of these statistical methods helps in making data-driven decisions by assessing the likelihood of different outcomes.

Random sampling is a technique where each member of a population has an equal chance of being selected. This ensures that the sample is unbiased and represents the broader population.
In our problem, a random sample of 3 individuals is chosen from a group of 10. This use of random sampling allows us to make probabilistic claims about the preferences for football and basketball, assuming the sample is representative of the larger group.
Random sampling is crucial in drawing accurate inferences because it minimizes selection bias, ensuring every possible group of 3 individuals from the total group of 10 has the same chance of being part of the sample. This methodology helps in enhancing the reliability of the conclusions drawn from the sample data.