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

Answer:

Age    |    Believe   |   Not believe  |   Total

<45     |    0.148      |        0.422       |   0.570

>45     |    0.301      |        0.129        |   0.430

Step-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:

$0.51=P(B;<45)*0.43+0.70*0.57\\\\P(B;<45)=\frac{0.51-0.70*0.57}{0.43} =\frac{0.11}{0.43}= 0.26$

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