# Mathematics of Banner Ads: Visitor Loyalty and CTR

First of all: why should a website’s visitor loyalty have any effect at all on the CTR we can expect to achieve with a banner ad? What does the one have to do with the other? To understand the connection, let’s take a look at an overly simplistic example. Suppose we place a banner ad on a website and get in total 3 impressions (granted, not a realistic number, but I’m only trying to make a point here). From previous campaigns we know that a visitor clicks on our ad with a probability of 0.1 = 10 % (which is also quite unrealistic).

The expected number of clicks from these 3 impressions is …

… 0.1 + 0.1 + 0.1 = 0.3 when all impressions come from different visitors.

… 1 – 0.9^3 = 0.27 when all impressions come from only one visitor.

(the symbol ^ stands for “to the power of”)

This demonstrates that we can expect more clicks if the website’s visitor loyalty is low, which might seem counter-intuitive at first. But the great thing about mathematics is that it cuts through bullshit better than the sharpest knife ever could. Math doesn’t lie. Let’s develop a model to show that a higher vistor loyalty translates into a lower CTR.

Suppose we got a number of I impressions on the banner ad in total. We’ll denote the percentage of visitors that contributed …

… only one impression by f(1)
… two impressions by f(2)
… three impressions by f(3)

And so on. Note that this distribution f(n) must satisfy the condition ∑[n] n·f(n) = I for it all to check out. The symbol ∑[n] stands for the sum over all n.

We’ll assume that the probability of a visitor clicking on the ad is q. The probability that this visitor clicks on the ad at least once during n visits is just: p(n) = 1 – (1 – q)^n (to understand why you have the know about the multiplication rule of statistics – if you’re not familiar with it, my ebook “Statistical Snacks” is a good place to start).

Let’s count the expected number of clicks for the I impressions. Visitors …

… contributing only one impression give rise to c(1) = p(1) + p(1) + … [f(1)·I addends in total] = p(1)·f(1)·I clicks

… contributing two impressions give rise to c(2) = p(2) + p(2) + … [f(2)·I/2 addends in total] = p(2)·f(2)·I/2 clicks

… contributing three impressions give rise to c(3) = p(3) + p(3) + … [f(3)·I/3 addends in total] = p(3)·f(3)·I/3 clicks

And so on. So the total number of clicks we can expect is: c = ∑[n] p(n)·f(n)/n·I. Since the CTR is just clicks divided by impressions, we finally get this beautiful formula:

CTR = ∑[n] p(n)·f(n)/n

The expression p(n)/n decreases as n increases. So a higher visitor loyalty (which mathematically means that f(n) has a relatively high value for n greater than one) translates into a lower CTR. One final conclusion: the formula can also tell us a bit about how the CTR develops during a campaign. If a website has no loyal visitors, the CTR will remain at a constant level, while for websites with a lot of loyal visitors, the CTR will decrease over time.