Final answer:
To find the probability that you must wait more than 8 hours for the first order to come in, we can use the Poisson distribution and the exponential distribution. The probability is approximately 99.9665%.
Step-by-step explanation:
To find the probability that you must wait more than 8 hours for the first order to come in, we can use the Poisson distribution. The Poisson distribution can be used to model the number of events occurring in a fixed interval of time, given the average rate of events per time period.
In this case, the average frequency of orders per day is 6. Since we are interested in the time it takes for the first order to come in, we can assume that the time between orders follows an exponential distribution. The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process.
The exponential distribution has a parameter called lambda (λ), which represents the average number of events per unit of time. In this case, λ is equal to 6 orders per day. The probability that you must wait more than 8 hours for the first order to come in can be found by calculating the cumulative distribution function (CDF) of the exponential distribution.
To calculate this probability, we can use the formula: P(x > 8) = 1 - P(x ≤ 8), where x is the time between orders.
For the exponential distribution, the formula for the CDF is: P(x ≤ t) = 1 - e^(-λt), where t is the desired time period.
Plugging in the values, we get: P(x > 8) = 1 - (1 - e^(-6*8)) ≈ 1 - 0.000335 ≈ 0.999665.
Therefore, the probability that you must wait more than 8 hours for the first order to come in is approximately 0.999665, or 99.9665%.