I run a small business, and I'm keen on understanding my market a little more. Market surveying and similar tools are not available to me, mainly due to time and budget. I don't require rock-solid data, just something that allows me to add a "pinch of salt" to my reasoning once in a while.

Is it appropriate to use Bayes' theorem to determine belief that my customers will increase as the market (population) increases, even as my competition increases? This assumes homogenous geographic distribution of customer populations and competition, which is inaccurate, and assumes that all customers are drawn equally to all businesses, which is again (probably) inaccurate. But at least it gives me a starting point beyond my ill-informed hunches.

I am not a statistician, which no doubt will be very evident, and have encountered several questions while trying to make sense of this approach. Apologies in advance for the number of questions here.

In my limited understanding, Bayes' theorem might help me modulate my beliefs regarding my competition, the market place, and so on.

I guess my first question should be: is this an appropriate use of Bayes' theorem?

Step 1: Constructing some formulae

Let's say:

$ P(C) = probability\ customer\ numbers\ will\ increase \\$ $ P(M) = probability\ market\ population\ will\ increase \\$ $ P(K) = probability\ competition\ will\ increase $

...then I presume I should be able to construct something like the following:

$ P(C|M) = \frac{P(M|C)P(C)}{P(M)} $

and

$ P(C|K) = \frac{P(K|C)P(C)}{P(K)} $

Second question: am I doing it right? :)

Step 2: Using appropriate data

The numbers I have are as follows:

• Professionals in my field, in the UK: approx 4,500.
• Growth in my field: approx 1,000 per year.
• Shrinkage in my field (due to retirement etc): approx 500 per year.
• Number of people in my city, and the rate at which it's growing.
• Percentage of a typical population that uses my profession's services: 10%.
• Month-on-month customer data in my business.

I can probably get more precise figures regarding the number of professionals in my field.

Regarding customer data, I have 18 months of past figures in a spreadsheet. In 9 of 17 completed months, customer numbers are higher than the previous month. This suggests to me that:

$ P(C) = \frac{9}{17} = 0.53 $

Regarding the market (population), let's say local government projects growth from 1,000,000 to 1,100,000 in the next year. This suggests to me that:

$ P(M) = \frac{1100000}{1000000} = 1.1 $

Regarding the competition, I'd assume:

$ P(K) = \frac{4500 + 1000 - 500}{4500} = 1.\dot{3} $

Third question: The figures may be accurate, but am I using them appropriately?

Step 3: The results

$ P(C|M) = \frac{P(M|C) \cdot 0.53}{1.1} $

This is where I run out of steam. How do I determine $P(M|C)$, the probability of the market increasing given that my customers also increase? Since my business has negligible effect on market size should I simply consider that to be 100%, i.e. 1? My business will also have minimal effect on competition (or so I believe, although maybe new local businesses are deterred by its presence) so I'd assume that similarly $ P(K|C) = 1 $.

Fourth question: Is that a sensible approach? Somehow I suspect not.

If that approach is right, the answers become:

$ P(C|M) = \frac{1 \cdot 0.53}{1.1} = \frac{0.53}{1.1} = 0.4\dot{8}\dot{1} $

$ P(C|K) = \frac{1 \cdot 1.\dot{3}}{1.1} = \frac{1.\dot{3}}{1.1} = 1.\dot{2}\dot{1} $

Fifth question: How should I then combine these two beliefs that customer numbers will go up as the market increases but competition also increases? Since they're probabilities, can I simply multiply them? That would suggest:

$ 0.4\dot{8}\dot{1} \cdot 1.\dot{2}\dot{1} = 0.584 $

Many thanks for any help you can offer, and apologies again for asking a lot of questions. I'm perfectly happy to be pointed toward useful reading materials, although my statistics and mathematics abilities are (clearly) limited, so novice-friendly info would be appreciated.