10. The time required to repair a machine is an exponential random variable with rate $$ \lambda=0.5 $$ downs/hour. a) What is the probability that a repair time exceeds 2 hours? b) What is the probability that the repair time will take at least 4 hours given that the repair man has been working on the machine for 3 hours?

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**a)**  
The probability that the repair time exceeds 2 hours is given by the survival function of the exponential distribution:
$$
P(X > 2) = e^{-\lambda \cdot 2} = e^{-0.5 \cdot 2} = e^{-1}.
$$

**b)**  
The exponential distribution has the memoryless property. This means that:
$$
P(X > 4 \mid X > 3) = P(X > 4 - 3) = P(X > 1) = e^{-\lambda \cdot 1} = e^{-0.5}.
$$
**a)** The probability that the repair time exceeds 2 hours is $$ e^{-1} $$. **b)** The probability that the repair time takes at least 4 hours given that it has already been working for 3 hours is $$ e^{-0.5} $$.

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