Let me try to take a stab at the lube case.
1. While it is rather random to use cars per 1,000 inhabitants as your metric, it is useful because that is the metric that the case has already established for Russia and Turkey. Table 1 shows information on Turkey, Russia and Germany, but omits the car density for Germany. Thus, it then becomes an exercise to try to estimate that missing information. Some clarifying questions are needed to figure out that "car density" means "# of cars per 1000 inhabitants". Once you know that is what you are solving, then it is just an estimation exercise. In this case, having some sense of the German population (80M) would help, but if you don't, it is fine. You could just say, I am not too familiar with the population of Germany, I know that the US is about 300M, would it be fair to assume that Germany is about half the size of the US?" Your interviewer would probably then just give you the accurate information, not penalizing you for not knowing it but appreciating that you understand how to find the info. in this case 40M households and 1 care per household is the information the case suggests, in real life you could come up with any reasonable number and it would make sense, but your interviewer would probably then say something like, "that makes sense, but for this case let's just assume there are 40M households in Germany and on average 1 car per household". It doesn't make it necessarily true, just a consistent number for both you and the interviewer to ensure that you both can reach the same end conclusion.
So in the end, yea it is random, but that comfort with ambiguity is part of what they are testing.
2. Also, in choosing a pilot in Turkey. The case is intended to focus on 3 countries in particular that the company has already identified as preferred countries for expansion (Turkey, Germany and Russia). I think that is part of the clarification process, asking if there is any specific area or countries in Europe that they are considering expanding into. Once you get that information, it becomes clear that you are really only looking at 3 countries. And then you narrow it down even more by looking at the pros and cons of each of those countries. This leads you to focus on Turkey.
3. A pilot program is suggested in this case because data can only tell you so much, at some point you have to actually do it. And in this case it is best to do it on a small scale to ensure that it works. Yes data is essential, and that could be part of the risk/risk mitigation in the final recommendation, but in the end, consultants (especially Bain, as this is a Bain case) are partial to action. So the recommendation (especially for Bain) is generally preferred to be rooted in an actionable recommendation while considering the risks involved and next steps to address them.
Something like, "I would recommend the client gets started with a pilot program in Turkey for these three reasons, 1. blah, 2. blah, 3. blah., There are a few risks I would need to iron out first; primarily how valid the growth, profitability and customer demand projections we have are. In order to address this I would want to dive more analytically into the data and get a bit more market research. But with that said, I believe that a pilot program in Turkey is the best course of action based on the information available right now."
That's the best I can quickly think of addressing the issues you mentioned. It's probably important to mention that the solution that is recommended is by no means the only answer to the problem. If you had a different conclusion, it very well could be just as valid.
