Question

Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ=0.1. What isthe conditional probability that a repair takes at least 10 hours, given that it takes more than 8 hours?

Answer 1

The conditional probability that a repair takes at least 10 hours, given that it takes more than 8 hours, is approximately 54.69%.

Step-by-step explanation:

To find the conditional probability that a repair takes at least 10 hours, given that it takes more than 8 hours, we can use the exponential distribution. In this case, the parameter λ=0.1. Let's denote the event A as the repair taking at least 10 hours, and event B as the repair taking more than 8 hours. We want to find P(A|B), which represents the conditional probability of event A given event B.

To find P(A|B), we can use the formula for conditional probability:
P(A|B) = P(A ∩ B) / P(B)

In this case, P(A ∩ B) represents the probability that the repair takes at least 10 hours AND more than 8 hours. Since the repair cannot take less than 8 hours and then suddenly take more than 10 hours, P(A ∩ B) is equal to P(A), which is the probability that the repair takes at least 10 hours. Therefore, P(A|B) = P(A) / P(B).

Using the exponential distribution, we can calculate:
P(A) = 1 - P(X ≤ 10) = 1 - e^(-λx) = 1 - e^(-0.1*10) = 1 - e^(-1)

For P(B), we can use the same formula:
P(B) = 1 - P(X ≤ 8) = 1 - e^(-λx) = 1 - e^(-0.1*8) = 1 - e^(-0.8)

Now, we can substitute these values back into the formula for conditional probability:
P(A|B) = (1 - e^(-1)) / (1 - e^(-0.8))

Using a calculator, we can find that P(A|B) is approximately 0.5469, or 54.69%.