SOLUTION: The studies of a single machine tool system showed that the machine operates before breaking down is exponentially distributed with a mean of 10 hours. a) Find the probabilit

Question 1192426: The studies of a single machine tool system showed that the machine
operates before breaking down is exponentially distributed with a mean of
10 hours.
a) Find the probability that the machine operates at least 12 hours before
breaking down.
b) Find the probability that the machine operates at least 14 hours but
less than 20s hours before breaking down.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
**a) Probability of operating at least 12 hours**
* **Given:**
* Mean time to failure (μ) = 10 hours
* Rate parameter (λ) = 1/μ = 1/10 = 0.1 per hour
* **Probability of operating at least 12 hours:**
* P(X ≥ 12) = 1 - P(X < 12)
* P(X ≥ 12) = 1 - (1 - e^(-λx))
* P(X ≥ 12) = 1 - (1 - e^(-0.1 * 12))
* P(X ≥ 12) = 1 - (1 - e^(-1.2))
* P(X ≥ 12) = e^(-1.2)
* P(X ≥ 12) ≈ 0.3012
**b) Probability of operating between 14 and 20 hours**
* **Probability of operating at least 14 hours:**
* P(X ≥ 14) = e^(-0.1 * 14) = e^(-1.4) ≈ 0.2466
* **Probability of operating at least 20 hours:**
* P(X ≥ 20) = e^(-0.1 * 20) = e^(-2) ≈ 0.1353
* **Probability of operating between 14 and 20 hours:**
* P(14 ≤ X < 20) = P(X ≥ 14) - P(X ≥ 20)
* P(14 ≤ X < 20) = 0.2466 - 0.1353
* P(14 ≤ X < 20) ≈ 0.1113
**Therefore:**
* a) The probability that the machine operates at least 12 hours before breaking down is approximately 0.3012.
* b) The probability that the machine operates at least 14 hours but less than 20 hours before breaking down is approximately 0.1113.

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