Final answer:
The expected time until the second failure of the laser is 50,000 hours. Probability cannot be calculated without additional details regarding the specific event or timeframe being considered.
Step-by-step explanation:
The question relates to the exponential distribution, which is a common model used in reliability engineering and for describing time until failure. Given a mean time to failure of 25,000 hours for a laser in a cytogenics machine, we can characterize the distribution of these failure times using the exponential distribution's probability density function (PDF) and cumulative distribution function (CDF).
To find the expected time until the second failure, we use the property of the exponential distribution that the sum of n exponentially distributed random variables is a gamma distribution with parameters n and the rate of the original exponential distribution. In this case, because the mean (μ) is 25,000 hours, the rate (λ) is 1/μ, or 1/25,000 per hour. The expected time until the second failure is then simply 2*μ, as the mean of the gamma distribution with these parameters is n/λ.
Part A: The expected time until the second failure is twice the mean time until a single failure, so it's 2 * 25,000 hours = 50,000 hours.
Part B: The probability question seems incomplete as the student did not specify what event we are to find the probability of. If they intended to ask for the probability of the second failure occurring within a certain timeframe, this can be found using the CDF of the gamma distribution with the appropriate parameters. If more details are given, a more specific probability can be calculated.