In a survey of 2000 adults 50 years and older of whom 40% were retired and 60% were pre-retired, the following question was asked: Do you expect your income needs to vary from year to year in retirement? Of those who were retired, 33% answered no, and 67% answered yes. Of those who were pre-retired, 32% answered no, and 68% answered yes. If a respondent in the survey was selected at random and had answered yes to the question, what is the probability that he or she was retired? (Round your answer to three decimal places).

Question

In a survey of 2000 adults 50 years and older of whom 40% were retired and 60% were pre-retired, the following question was asked: Do you expect your income needs to vary from year to year in retirement? Of those who were retired, 33% answered no, and 67% answered yes. Of those who were pre-retired, 32% answered no, and 68% answered yes. If a respondent in the survey was selected at random and had answered yes to the question, what is the probability that he or she was retired? (Round your answer to three decimal places).

Answer 1

To find the probability that a respondent who answered "yes" to the question is retired, we need to use conditional probability.

First, let's calculate the probability of a respondent being retired given that they answered "yes".

From the information given, the percentage of retired respondents who answered "yes" is 67%, and the percentage of pre-retired respondents who answered "yes" is 68%.

To calculate the probability that a respondent was retired given that they answered "yes", we divide the number of retired respondents who answered "yes" by the total number of respondents who answered "yes":

Total number of retired respondents who answered "yes" = 40% (proportion of retired respondents) * 67% (percentage of retired respondents who answered "yes") = 0.40 * 0.67 = 0.268

Total number of pre-retired respondents who answered "yes" = 60% (proportion of pre-retired respondents) * 68% (percentage of pre-retired respondents who answered "yes") = 0.60 * 0.68 = 0.408

Total number of respondents who answered "yes" = Total number of retired respondents who answered "yes" + Total number of pre-retired respondents who answered "yes" = 0.268 + 0.408 = 0.676

Therefore, the probability that a respondent who answered "yes" is retired is:

Probability of being retired given "yes" = (Total number of retired respondents who answered "yes") / (Total number of respondents who answered "yes") = 0.268 / 0.676 ≈ 0.396.

Therefore, the probability that a respondent who answered "yes" is retired is approximately 0.396 (rounded to three decimal places).