Final answer:
The student's question concerning the probability of receiving no calls within certain time intervals is solved using the exponential distribution's properties. By calculating the probability for a single interval and then raising it to the power of three, due to the independence of the intervals, we find the overall probability for no calls in all periods.
Step-by-step explanation:
The subject matter of this question falls within the scope of Mathematics, specifically within the study of probability theory and exponential distribution. The task is to compute the probability that no calls occur during three separate 5-minute intervals, given an average rate of calls that fits an exponential distribution with a mean time between calls of 10 minutes. To address this, first, we need to determine the rate (lambda, λ) of the exponential distribution, which is the reciprocal of the mean (so λ = 1/10 per minute).
The probability that no calls are received within a specific time interval (t) for an exponential distribution is given by the formula P(X > t) = e^(-λ*t). For each 5-minute interval (which can be expressed as t = 1/12 hours or t = 5/60 minutes), the probability that no calls are received is P(X > 5/60) = e^(-(1/10)*(5/60)). Since the intervals are independent, the combined probability is the product of the individuals:
P(no calls in all intervals) = (e^(-(1/10)*(5/60)))^3. Executing the computation with λ = 1/10 and t = 5/60 yields the probability for each interval. Multiplying that probability by itself three times results in the final answer to the student's question.