A company has recently started building a new plant. Letting A represent the number of months to complete the project, the probability distribution of A is:

` $$ P(A)=\frac{A}{22}, A = 4,5,6,7. $$ `

i) Show that P(A) is a probability function.

ii) What is the probability that the plant will be completed in at most 6 months?

iii) What is the probability it will be completed in at least 5 and not more than 6 months?

#### Solution

Given that the probability distribution of A is

` $$ P(A)=\frac{A}{22}, A = 4,5,6,7. $$ `

i) `$P(A=4) = \frac{4}{22}$`

, `$P(A=5)=\frac{5}{22}$`

, `$P(A=5)=\frac{6}{22}$`

and `$P(A=7)=\frac{7}{22}$`

.

All `$P(A)> 0$`

for `$A=4,5,6,7$`

and the total probability is

` $$ \begin{aligned} & \sum_{A=4}^{7} P(A)\\ &=\frac{4}{22}+\frac{5}{22}+\frac{6}{22}+\frac{7}{22}\\ &=\frac{22}{22}\\ &=1. \end{aligned} $$ `

Thus $P(A)$ is a probability function.

ii) The probability that the plant will be completed in at most 6 months is

` $$ \begin{aligned} P(A<=6) & = P(A=4)+P(A=5)+P(A=6)\\ &= \frac{4}{22}+\frac{5}{22}+\frac{6}{22}\\ &=\frac{15}{22}\\ &=0.6818182 \end{aligned} $$ `

iii) The probability that it will be completed in at least 5 and not more than 6 months is

` $$ \begin{aligned} P(A>=5 \text{ and } A =6) & = P(A=5)+P(A=6)\\ &= \frac{5}{22}+\frac{6}{22}\\ &=\frac{11}{22}\\ &=0.5 \end{aligned} $$ `

#### Further Reading

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