A few years ago, a survey commissioned by The World Almanac and Maturity News Service reported that 51% of the respondents did not believe the Social Security system will be secure in 20 years. Of the respondents who were age 45 or older, 70% believed the system will be secure in 20 years. Of the people surveyed, 57% were under age 45. One respondent is selected randomly. a. What is the probability that the person is age 45 or older? b. What is the probability that the person is younger than age 45 and believes that the Social Security system will be secure in 20 years? c. If the person selected believes the Social Security system will be secure in 20 years, what is the probability that the person is 45 years old or older? d. What is the probability that the person is younger than age 45 or believes the Social Security system will not be secure in 20 years?
Accepted Solution
A:
Answer:Age | Believe | Not believe | Total<45 | 0.148 | 0.422 | 0.570>45 | 0.301 | 0.129 | 0.430Step-by-step explanation:We have to construct a probability matrix for this problem.Of the people surveyed, 57% were under age 45. That means that 43% is over age 45.70% of the ones who were 45 or older, believe the Social Security system will be secure in 20 years.The Believe proportion is 51%.Then, the proportion that believe and are under age 45 is:[tex]0.51=P(B;<45)*0.43+0.70*0.57\\\\P(B;<45)=\frac{0.51-0.70*0.57}{0.43} =\frac{0.11}{0.43}= 0.26[/tex] We can now construct the probability matrix for one respondant selected randomly:[tex]P(<45\&B)=0.57*0.26=0.148\\\\P(<45\&NB)=0.57*(1-0.26)=0.57*0.74=0.4218\\\\P(>45\&B)=0.43*0.7=0.301\\\\P(>45\&NB)=0.43*(1-0.7)=0.43=0.3=0.129[/tex]Age | Believe | Not believe | Total<45 | 0.148 | 0.422 | 0.570>45 | 0.301 | 0.129 | 0.430