The probability that first processing machine is not working properly at any time is 0.07, and the probability that the second machine is not working properly at any time is 0.07 and the probability that third machine is not working properly is 0.09.

Find the probability that at least one machine will be working properly at any given time. Provide answer to four decimal precision

#### Solution

Let $A$ be the event that first processing machine is not working properly at any time.

Let $B$ be the event that second processing machine is not working properly at any time.

Let $C$ be the event that third processing machine is not working properly at any time.

Given that $P(A) = 0.07$, $P(B)=0.07$ and $P(C) = 0.09$.

Let $E$ be the event that at least one machine will be working properly at any given time.

Then the probability that at least one machine will be working properly at any given time is

` $$ \begin{aligned} P(E) & = 1- P(\text{no machine working properly})\\ & = 1- P(A\cap B\cap C)\\ &\quad \quad \text{( A, B, C are independent)}\\ &= 1- P(A)P(B)P(C)\\ &= 1- (0.07)*(0.07)*(0.09)\\ &= 1- 0.000441\\ &= 0.999559 \end{aligned} $$ `

#### Further Reading

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