How to statistically measure the potential effectiveness of TV ads (based on customer survey data)? My statistical knowledge is rusty, so I decided to ask this question about an analysis: let's say we have conducted a survey on our customers and have asked them how often they have watched TV in the last week. It turns out that 55% of our customers have watched TV in the last week for at least one session. So, we can say:
$$
P(WatchTV | Customer) = 0.55
$$
Now, we would like to estimate how many of the TV viewers could be our potential customer/consumer of our product (to find out whether it's worth to buy TV ads). In other words, we would like to estimate our sales lead among TV viewers, i.e. $P(SalesLead|WatchTV)$. To find this out, we can use the Bayesian rule:
$$
P(SalesLead|WatchTV) = \frac{P(WatchTV|SalesLead)P(SalesLead)}{P(WatchTV)}
$$
Now, finding $P(WatchTV)$ is easy, as we can rely on an estimate of the proportion of the whole population who watch TV. Also, we can estimate $P(SalesLead)$ by considering the proportion of the population who potentially need our product/service and will use it (I know, this estimate might be prone to be biased or over-optimistic; but, for now, let's ignore this point).
But regarding the value of $P(WatchTV|SalesLead)$: is it correct to say that we can assume the behavior of our potential customers would be more or less the same as our current customers, especially if number of our current customers is relatively high? And then if that's true (to a good degree of precision), can we replace $P(WatchTV|SalesLead)$ by $P(WatchTV|Customer)$, implying the probability of watching TV in our current customers and potential customers is the same?

*

*How much erroneous is the above analysis?

*Or is there any other approach and/or survey question(s) that we can ask our current customers that helps us to better estimate the value of $P(SalesLead|WatchTV)$?

*What about the demographics data (age group, gender, tech usage, etc.)? i.e. how can we include that data in our analysis, since the demographics of TV viewers might (significantly) differ from those of our current/potential customers and the above analysis, if correct, does not explicitly model it?


Assuming that the behavior of potential customers is similar to current customers would be erroneous in most of the cases.
For example, suppose your products are high-end TVs. Then, $P(WatchTV|Customer)$ would likely be higher compared to $P(WatchTV|SalesLead)$. On the other hand, if your products are, say, high-end gaming consoles, then $P(WatchTV|Customer)$ would likely be lower compared to $P(WatchTV|SalesLead)$. Note that $SalesLead$, or potential customers, also contain individuals who may not become customers eventually. It is this collection of individuals which makes the difference between $P(WatchTV|Customer)$ and $P(WatchTV|SalesLead)$. The following Venn diagram illustrates this point. Here, the overall rectangle represents the whole population (shall be denoted as $U$), the light yellow region represents the sales leads (to be denoted as $A$), the dark yellow represents the customers (denoted as $B$) and the translucent blue region represents the people who watch TV (denoted as $C$). Here, $P(WatchTV|SalesLead)$ becomes $P(A\cap C) / P(A)$, and $P(WatchTV|Customer)$ reduces to $P(B\cap C) / P(B)$. A priori, there is no relation between $P(A\cap C) / P(A)$ and and $P(B\cap C) / P(B)$. 
On the other hand, $P(WatchTV|SalesLead) \simeq P(WatchTV|Customer)$ would be an indication that watching TV ads do not influence a potential customer to become an actual customer.
Estimation of $P(SalesLead|WatchTV)$ is required to be done in some other way. Since $P(SalesLead|WatchTV) = P(SalesLead\cap WatchTV) / P(WatchTV)$, $P(SalesLead\cap WatchTV)$ can be estimated from other demographic information collected in a survey to estimate $P(WatchTV)$ and $P(SalesLead)$. The demographics data is used to properly estimate $P(SalesLead)$, and an additional information to be collected in the survey can be the size of the group of people in the $SalesLead$ category who watch TV.
However, more useful to the actual purpose, which is to determine the utility of buying TV ads in converting sales leads to actual customers, would be to conduct surveys within customers on whether earlier TV ads have influenced their purchase decision.
