A telephone survey conducted by the Maritz Marketing Research company found that 43% of Americans expect to save more money next year than they saved last year. Forty-five percent of those surveyed plan to reduce debt next year. Of those who expect to save more money next year, 81% plan to reduce debt next year. An American is selected randomly. a. What is the probability that this person expects to save more money next year and plans to reduce debt next year? b. What is the probability that this person expects to save more money next year or plans to reduce debt next year? c. What is the probability that this person expects to save more money next year and does not plan to reduce debt next year? d. What is the probability that this person does not expect to save more money given that he/she does plan to reduce debt next year?

A) P(A⋂B) = 0.35

B) P(A⋃B)= 0.53

C) P(A⋂B′) = 0.08

D) P(A|B) = 0.778

Step-by-step explanation:

We know the following from the question:

- Let Proportion of Americans who expect to save more money next year than they saved last year be

P(A) and its = 0.43

-Let proportion who plan to reduce debt next year be P(B) and it's =0.81

A) probability that this person expects to save more money next year and plans to reduce debt next year which is; P(A⋂B) = 0.43 x 0.81 = 0.348 approximately 0.35

B) probability that this person expects to save more money next year or plans to reduce debt next year which is;

P(A⋃B)= P(A) + P(B) − P(A⋂B)

So, P(A⋃B)= 0.43 + 0.45 − 0.35 = 0.53

C). Probability that this person expects to save more money next year and does not plan to reduce debt next year which is;

P(A⋂B′) = P(A) − P(A⋂B)

P(A⋂B′) =0.43 − 0.35 = 0.08

D) Probability that this person does not expect to save more money given that he/she does plan to reduce debt next year which is;

P(A|B) = [P(A⋂B)] / P(B)

So P(A|B) =0.35/0.45 = 0.778

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt