Final answer:
To find the probability of an industrial robot not breaking down during the day, we use the Poisson distribution formula with λ = 0.5 for zero breakdowns. For the robot to work at least 4 hours without breaking down, we use the exponential distribution with λ = 0.25 for a 4-hour period.
Step-by-step explanation:
The breakdowns of an industrial robot are described by a Poisson distribution with an average rate of 0.5 breakdowns per 8-hour workday. This information will help us calculate the probabilities of different events related to the robot's operation.
Event A: Robot not breaking down during the day
To find the probability that the robot will not break down during the day, we use the Poisson probability formula:
P(X = k) = (e-λ * λk) / k!
For k = 0 breakdowns, and λ (average rate) = 0.5, we have:
P(X = 0) = (e-0.5 * 0.50) / 0! = e-0.5.
Event B: Robot working at least 4 hours without breaking down
To find the probability of the robot working for at least 4 hours without breaking down, we use the exponential distribution, which is related to the Poisson distribution:
P(X > t) = e-λt
For t = 4 hours (half the period of our Poisson distribution), we need to adjust λ to reflect the 4-hour period, so λ = 0.25. Then we have:
P(X > 4) = e-0.25*4 = e-1.
Thus, the probability of the industrial robot not breaking down during the 8-hour day is e-0.5, and the probability of it working for at least 4 hours without breaking down is e-1.