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Answer :
The average number of students at the coffee urn is 3, the expected time to get a cup of coffee is 15 seconds, the percentage of time the urn is being used is approximately 6.78%, the probability of three or more students being in the café can be calculated using the Poisson distribution.
a. To determine the average number of students at the coffee urn, we can use the concept of the Poisson distribution. The arrival rate of students is given as three per minute, which means the average number of students arriving in one minute is λ = 3. The Poisson distribution's mean is equal to its parameter λ, so the average number of students at the coffee urn is also 3.
b. The time taken by students to serve themselves follows an exponential distribution with a rate parameter of λ = 1/15 (since the average time is the reciprocal of the rate). The expected time to get a cup of coffee can be calculated as the reciprocal of the rate parameter, which is 15 seconds.
c. The percentage of time the urn is being used can be calculated by considering the time when at least one student is serving themselves. This can be expressed as 1 minus the probability that the urn is idle. The probability of the urn being idle can be calculated using the exponential distribution.
The rate parameter for the exponential distribution is λ = 1/15, so the probability of the urn being idle is e^(-λ), which is e^(-1/15) ≈ 0.9322. Therefore, the percentage of time the urn is being used is approximately 100% - 93.22% = 6.78%.
d. To calculate the probability that three or more students are in the café, we can use the Poisson distribution. The parameter λ is given as 3, so the probability of having exactly two students can be calculated using the Poisson probability mass function. Then, we can subtract this probability from 1 to get the probability of three or more students.
e. If the café installs an automatic vendor that dispenses a cup of coffee at a constant time of 15 seconds, it would not change the answer to part a because the arrival rate of students remains the same.
However, it would change the answer to part b. With the automatic vendor, the time taken to get a cup of coffee would be constant at 15 seconds, as opposed to being exponentially distributed.
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