Final answer:
To find the probability that more than a minute elapses between calls in a call center, we can use the exponential distribution and the average time between calls. By subtracting the probability that the time between calls is less than or equal to 60 seconds from 1, we can find the desired probability.
Step-by-step explanation:
To find the probability that more than a minute elapses between calls, we need to use the exponential distribution. The exponential distribution is characterized by the parameter lambda (λ), which represents the average rate of events occurring per unit of time. In this case, the average time between calls is 30 seconds, so λ = 1/30 calls per second.
To find the probability that more than a minute elapses between calls, we need to find the probability that the time between calls is greater than 60 seconds. This can be done by subtracting the probability that the time between calls is less than or equal to 60 seconds from 1.
P(time > 60 seconds) = 1 - P(time <= 60 seconds)
To find P(time <= 60 seconds), we can use the exponential cumulative distribution function:
P(time <= t) = 1 - e^(-λt)
Substituting in the values, we get:
P(time <= 60 seconds) = 1 - e^(-1/30 * 60)
P(time <= 60 seconds) = 1 - e^(-2)
Now we can calculate P(time > 60 seconds):
P(time > 60 seconds) = 1 - P(time <= 60 seconds)
P(time > 60 seconds) = 1 - (1 - e^(-2))
P(time > 60 seconds) = e^(-2)