Everyday Statistics, with Eddie Davila

– [Instructor] Lines. They’re everywhere. The lines we see at coffee shops and grocery stores and even those we can’t see like when we wait on hold at a call center. How long will I wait? What are the chances there won’t be line? The key to answering these questions is statistics and probability. Suppose a place of business has an average of six customers arrive per hour. So a customer arrives on average every 10 minutes. This business has a single employee that can help up to 10 customers per hour.

On average, it takes this employee six minutes to help each customer. Those two numbers are almost all that is needed to work through the most basic waiting line problems. Now that we know that customers arrive six per hour and we can help 10 per hour, we know the percentage of time our server is busy helping customers. This worker is busy 60% of the time. So one worker is busy 60% of the time. If we hire a second worker, each of these two workers will be busy 30% of the time.

Any guesses how busy the servers would be in a three server scenario? If you said 15%, nice try but perhaps rethink it. One server was busy 60% of the time. Two servers split up the work evenly so each was busy 30% of the time. So when we hire three workers to do the work of one worker, we’re splitting up 60% among three people. The servers are busy only 20% of the time.

None of those workers look very busy though. Why would we even hire a second worker and pay them to do so little? Well, let’s say our goal is to have customers wait in line less than 90% of the time. In other words, we want 90% of our customers served immediately. What are the conditions required to have no line when we have one server? The place must be empty. If we have even one customer in our store, the next person will get in line.

Trust me when I say the probability of an empty store in this situation is 40%. So 40% of the time, we have no line. In a two server scenario, things change. Now we can have an empty store or we can have one customer. Why? Because if we have one customer, we have one busy server and one server that is free. The next person to arrive will not need to wait. So we add the probability of the empty store scenario and the probability of the one customer scenario.

The empty store probability is 54%. The one customer probability is 32%. That’s an 86% probability. You may wonder why the empty store probability went up from 40% with one server to 54%. Two servers means customers get in an out faster so there’s a higher probability that the store is empty. How about if we add a third server? Now we have three no line solutions.

Empty store, one customer or even two customers all leave us with at least one free server. The empty store probability is 55%. The one customer probability is 33% and the two customer probability is 10%. If you add them up, you’ll find we now have a 98% probability. So should we hire one, two or three servers? One server would probably result in too many lines.

Two servers is much better but double the wages. Three servers is a little better than two servers and you’ll need to pay three workers. Which one would you choose? Well, it’s your business. Statistics and probability might not give you the final answer but they certainly make your choices much more clear. Get a firm foundation in statistics and probability and use your skills to help society wait in fewer lines without breaking your own bank.