Final answer:
To find the probability of the next customer arriving in less than one minute, we need to calculate the cumulative distribution function (CDF) of the inter-arrival time. Since the customers arrive randomly every 1 to 8 minutes, the inter-arrival time follows a Uniform distribution. The probability is calculated using the CDF equation.
Step-by-step explanation:
To find the probability that it takes less than one minute for the next customer to arrive after a customer arrives, you need to calculate the cumulative distribution function (CDF) of the inter-arrival time.
Since the customers arrive randomly every 1 to 8 minutes, the inter-arrival time follows a Uniform distribution.
The CDF of a Uniform distribution is given by:
CDF(x) = (x - a) / (b - a)
where x is the time, a is the minimum time, and b is the maximum time. In this case, a = 1 and b = 8.
Substituting these values into the equation, we get:
CDF(x) = (x - 1) / (8 - 1) = (x - 1) / 7
To find the probability that it takes less than one minute for the next customer to arrive, we need to calculate CDF(1). Substituting x = 1, we get:
CDF(1) = (1 - 1) / 7 = 0/7 = 0
Therefore, the probability that it takes less than one minute for the next customer to arrive is 0.