The Table Shows The Results Of A Survey In
Which People Were Asked Abo
The table shows the results of a survey in which people were asked about their marital and education status. (a) Find the probability that a randomly selected person is divorced given that they graduated from college. (b) Find the probability that a randomly selected person graduated from high school and is married. (c) Find the probability that a randomly selected person has less than a high school education or is single. (d) Are the events “graduating from high school” and “divorce” independent or dependent events?
Paper For Above instruction
In analyzing survey data related to marital and educational status, probability theory provides essential tools for understanding relationships between different demographic factors. The given problem involves calculating specific conditional and joint probabilities, as well as determining the independence of events based on survey results. This paper explores each part of the problem in detail, illustrating the applications of probability in social science research.
Part A: Probability of Divorce Given Graduation from College
To find the probability that a person is divorced given that they graduated from college, we employ conditional probability. If P(D|C) denotes the probability of being divorced given a college degree, then:
P(D|C) = P(D ∩ C) / P(C)
Where P(D ∩ C) is the probability that a person is both divorced and has a college degree, and P(C) is the probability that a person has a college degree. To compute these, data from the survey table are required. Assuming the total number of respondents (N), the number of college graduates, the number of divorced college graduates, these probabilities are calculated as ratios of the relevant counts over N.
For example, if the survey data indicates 200 respondents out of 1000 are college graduates, and among them, 50 are divorced, then:
P(D|C) = 50 / 200 = 0.25
This indicates a 25% chance that a college graduate is divorced, based on this hypothetical data. The actual calculation depends on the specific counts from the survey table.
Part B: Probability that a Person Graduated from High School and is Married
This part involves calculating the joint probability P(H ∩ M), where H indicates graduation from high
school, and M indicates being married. It is determined by dividing the number of respondents with both characteristics by the total number of respondents:
P(H ∩ M) = Number of respondents who graduated high school and are married / Total respondents
If the survey indicates that 300 people graduated from high school, and among them, 180 are married, then:
P(H ∩ M) = 180 / 1000 = 0.18
This reflects the probability that a randomly selected individual has both educational and marital status as specified.
Part C: Probability of Less than High School Education or Being Single
The probability that a randomly selected person has less than a high school education or is single involves calculating P(L ∪ S), where L represents less than high school education, and S represents being single.
Using the principle of inclusion-exclusion:
P(L ∪ S) = P(L) + P(S) - P(L ∩ S)
Suppose 150 respondents have less than high school education, 400 are single, and 50 are both less than high school educated and single. Then:
P(L ∪ S) = (150/1000) + (400/1000) - (50/1000) = 0.15 + 0.40 - 0.05 = 0.50
This calculation shows that there's a 50% chance that a person either has less than high school education or is single.
Part D: Independence of the Events “Graduating from High School” and “Divorce”
Two events are considered independent if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if:
P(A ∩ B) = P(A) * P(B)
To test this with the given events, we compare the observed joint probability of graduating high school and being divorced with the product of their individual probabilities.
If, for example, P(H) = 0.30 and P(D) = 0.10, then:
Product = 0.30 * 0.10 = 0.03
If the actual joint probability P(H ∩ D) from the data is significantly different from 0.03, then the events are dependent. If roughly equal, they are independent.
In most social settings, education level and marital status often influence each other due to socio-economic factors, suggesting potential dependence. Empirical data from the survey provides the definitive evidence to confirm this relationship.
In conclusion, probability calculations such as these are vital for understanding social phenomena and the interdependencies among demographic variables. Accurate interpretation of survey data requires careful analysis of the probabilities and their relationships, which can inform policy-making, targeted interventions, and further research in social sciences.
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