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 neither expects to save more money next year nor plans to reduce debt next year?

d. What is the probability that this person expects to save more money next year and does not plan to reduce debt next year?

### Solution

Let $M$ = expect to save more, $R$ = expect to reduce debt, $NM$ = don't expect to save more,

$NR$ = don't expect to reduce debt

Given that

$P(M)= 0.43$, $P(R)=0.45$, $P(RM) = 0.81$

$P(NR|M)=1- P(R|M) = 1-0.81 = 0.19$

$P(NM) = 1- P(M) = 1- 0.43 = 0.57$

$P(NR) = 1- P(R) = 1- 0.45 = 0.55$

a. The probability that this person expects to save more money next year and plans to reduce debt next year is

` $$ \begin{aligned} P(M\cap R) & = P(M) * P(RM)\\ &= 0.43*0.81\\ &= 0.3483 \end{aligned} $$ `

b. The probability that this person expects to save more money next year or plans to reduce debt next year is

` $$ \begin{aligned} P(M\cup R) &= P(M) + P(R) - P(M\cap R)\\ & = 0.43+0.45 - 0.3483\\ & = 0.5317 \end{aligned} $$ `

c. The probability that this person neither expects to save more money next year nor plans to reduce debt next year is

` $$ \begin{aligned} P(\text{neither save nor reduce debt}) & = 1- P(M\cup R)\\ & = 1- 0.5317\\ & = 0.4683 \end{aligned} $$ `

d. The probability that this person expects to save more money next year and does not plan to reduce debt next year is

` $$ \begin{aligned} P(M \cap NR) & = P(M)* P(NR|M)\\ & = 0.43 * 0.19\\ & = 0.0817 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators