Question

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.Construct a probability matrix for this problem.

Answer 1

Age | Believe | Not believe | Total

<45 | 0.148 | 0.422 | 0.570

>45 | 0.301 | 0.129 | 0.430

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:

`0.51=P(B;<45)*0.43+0.70*0.57\\\\P(B;<45)=(0.51-0.70*0.57)/(0.43) =(0.11)/(0.43)= 0.26`

We can now construct the probability matrix for one respondant selected randomly:

`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`

Age | Believe | Not believe | Total

<45 | 0.148 | 0.422 | 0.570

>45 | 0.301 | 0.129 | 0.430