A Small Town Has 5600 Residents The Residents In The Town We ✓ Solved

1 A Small Town Has 5600 Residents The Residents In The Town Were Ask

Analyze a scenario involving a small town with 5600 residents where individuals were asked whether or not they favored building a new bridge across the river. The data is broken down by gender and responses: in favor or opposed. Based on this information, you are asked to calculate various probabilities related to the residents’ opinions and demographics, evaluate the relationships between gender and opinions, and explore basic combinatorial and probability concepts.

Sample Paper For Above instruction

The scenario involves understanding the probability distribution of residents' opinions about building a bridge across the river, considering their gender. We are provided with data on the counts of men and women who are in favor or oppose the bridge. This information enables us to compute various probabilities, analyze the independence or mutual exclusivity of events, and relate the data to fundamental combinatorial and probability principles.

Understanding the Data and Probabilities

Suppose the breakdown of the 5600 residents is as follows: the number of men in favor, men opposed, women in favor, and women opposed. While actual counts are not provided in the prompt, a typical analysis involves calculating probabilities based on such data.

• Part A: The probability that a randomly selected resident favors the bridge is given by dividing the total number of residents in favor by 5600. Mathematically, P(In Favor) = Number in Favor / 5600.
• Part B: The probability that a randomly selected resident is a man and in favor equals the number of men in favor divided by 5600: P(Man and In Favor) = Number of Men in Favor / 5600.
• Part C: The probability that a resident is a man or in favor involves the union of these events: P(Man or In Favor) = P(Man) + P(In Favor) - P(Man and In Favor).
• Part D: The probability that a woman is in favor, given she is a woman, is a conditional probability: P(In Favor | Woman) = P(Woman and In Favor) / P(Woman).

Mutual Exclusivity and Independence

Mutual exclusivity between gender and opinion about the bridge typically does not hold because individuals can be both a woman and in favor or opposed. Therefore, these events are not mutually exclusive. To assess independence, we compare whether the probability of being in favor given gender equals the overall probability of being in favor: P(In Favor | Gender) = P(In Favor). If the equality holds, then the events are independent; otherwise, they are dependent.

Additional Probability and Combinatorial Concepts

• Part 2: The number of different combinations of selecting 5 students from a group of 9 students is computed using the combination formula: C(9, 5) = 9! / (5!(9-5)!).
• Part 3: The sample space for drawing one card from a standard deck comprises 52 equally likely outcomes, each representing a specific card (e.g., Ace of Spades, 2 of Hearts, etc.).

Conclusion

By understanding probabilities based on population data, assessing independence, and exploring basic combinatorial calculations, we obtain insights into the residents' opinions and demographics. Such analysis is fundamental in statistical inference, decision-making, and understanding population characteristics within a community.

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