The probability that house sales will increase in the next 6 months is estimated to be 0.25. The probability that the interest rates on housing loans will go up in the same period is estimated to be 0.74. The probability that house sales or interest rates will go up during the next 6 months is estimated to be 0.89.

a) What is the probability that both house sales and interest rates will increase during the next 6 months? Show or explain how you obtain your answer.

b) What is the probability that neither house sales nor interest rates will increase during the next 6 months? Show or explain how you obtain your answer.

c) What is the probability that house sales will increase but interest rates will not during the next 6 months? Show or explain how you obtain your answer.

#### Solution

Let $H$ denote the event that the house sales increase in the next 6 month. Let $I$ denote the event that the interest rates on housing loads go up in the same period.

Given that `$P(H) = 0.25$`

and `$P(I)=0.74$`

.

The probability that house sales or interest rates will go up during the next 6 months is estimated to be 0.89, i.e., `$P(H\text{ OR } I)= P(H\cup I) = 0.89$`

.

a. The probability that both house sales and interest rates will increase during the next 6 months is

` $$ \begin{aligned} P(H \text{ and } I) &= P(H\cap I)\\ &= P(H)+P(I) - P(H\cup I)\\ &= 0.25 +0.74-0.89\\ &= 0.1 \end{aligned} $$ `

b. The probability that neither house sales nor interest rates will increase during the next 6 months is same as the complementary of the probability that either house sales or interest rates will increase during the next 6 months.

` $$ \begin{aligned} P(\text{ neither }) &= 1- P(\text{ either })\\ &= 1- P(H\text{ OR } I)\\ &= 1- P(H\cup I)\\ &= 1-0.89\\ &= 0.11 \end{aligned} $$ `

c. The probability that house sales will increase but interest rates will not during the next 6 months is

` $$ \begin{aligned} P(\text{ Only H }) &= P(H)- P(H\cap I)\\ &= 0.25- 0.1\\ &= 0.15 \end{aligned} $$ `

#### Further Reading

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