Math: Likelihood of having to refill the water cooler

Factual Questions
1

Suppose 10 employees share a water cooler (call the employees e0 through e9). The water cooler gets its water from 10-gallon jugs. Suppose that whenever an employee comes to get water, and there is not enough water to fill his mug, he faithfully replaces the empty jug with a full jug, then fills his mug. Suppose further that the size of the employees’ mugs varies from 1 to 2 cups, and there is no way to add the sizes in combination to total 10-gallons exactly – that is, there is no special relationship between the size of the cups and the size of the jug (call the sizes s0 through s9). And suppose that the order in which employees make a trip to the water cooler is indeterministic, but that every employee goes to the water cooler a certain number of times per day (call these w0 through w9).

How can one express the probability that an employee will have to replace the jug? e.g. what is p0 in terms of s0-s9 and w0-w9?

Is it straight multiplication (e.g. p0=s0w0/(s0w0+s1w1+…+s9w9)) or is there something more complex going on here?

How does the problem change if instead of faithfully changing the jug, the employees drain the jug of all remaining water, and leave it for the next sap to replace?

Thanks.

2

If it’s anything like the coffee in my math department this is a highly inaccurate assumption.

/me grumbles about being the only one to ever make a new pot of coffee.

3

What’s p0 again?

4

er, the probability that e0 will have to refill the jug

5

I think you’re right. We are approximate the event (“jug runs out of water”) as a random event that can happen any second while someone is filling his/her mug. So the probability is proportional to the size of his/her mug and the frequency of refills.

So you want the probability that the next person to come to the jug is you. This is proportional to the number of times you visit the jug, so it’s w0/(w0+w1+…+w9).