Joe runs a small used auto business. On average he sells a car every two weeks and this provides Joe with a modest, but comfortable existence. The one thing Joe dreads is an 8 week period without business. Two months without income will flatten him, and having seen a six week business drought he is keen to get some idea of the likelihood of an eight week drought. So we helped him out.
In any single week the probability of a sale is 0.5 or 50%. We’ll denote a week with a sale by ‘S’ and a week with no sale by ‘N’. So for any given week we have two scenarios:
S – 50%
N – 50%
Simple. There are four scenarios for two weeks – SS, SN, NS, NN. Now we are going to make an assumption that will probably be largely true for Joe’s business. We will assume that each week is independent of every other week – meaning that the lack of a sale in one week does not mean a sale is more likely the following week. In this scenario we multiply the probabilities together and since S and N both have a probability of 0.5, each of the four scenarios has a probability of 0.25, or 25%. Another way of looking at this is to consider the combinations of possibilities – of which there are four, and each as likely as the other.
SS – 25%
SN – 25%
NS – 25%
NN – 25%
There is a 25% chance that Joe will not make a sale, and a 75% chance that he will make one sale or two over a two week period. You may already detect a pattern. For any number of weeks x, the number of combinations of sales is 2 multiplied by itself x times – or 2 to the power x. For eight weeks this is 256. There are 256 eight week patterns of sales that Joe can experience, and each is just as likely as the other. One of these patterns is NNNNNNNN. Yes Joe will experience an eight week period without sales approximately once every 5 years (256 weeks). He has been in business four years and will soon be living on borrowed time. When this news is given to Joe he decides he needs to cut back on the sleazy nightclubs and cigars.
What surprised and unsettled Joe was the news that NNNNNNNN is just as likely as any other pattern – SNSSNNSS for example. Did this mean that the SSSSSS run he had a few months ago was nothing more than a random fluke, and not down to the new ‘charm offensive’ sales technique he was using. We told him we didn’t know and only time would indicate whether his average sales per week was increasing – but there would always be uncertainty associated with any measurement of performance – it might be random and it might not.
We’ve kept things deliberately simple in this example, just to illustrate some points. Of course Joe might sell three cars in a given week – but that is a different problem. What we are trying to show here is that random variations in variables mislead us all the time. We are programmed to look for cause and effect and will very often attribute a cause to something that is completely random and without cause. Statisticians have ways of dealing with this, but guess what. They are uncertain too.
So the next time you experience a four week lull in business remember that it is most likely just the effects of randomness. Same with that three month surge in sales. If only you could find what caused it. There is a very good chance that nothing caused it, and was just randomness playing its games.
Below is a simple simulation of Joe’s eight week running sales – he did hit that zero.