Frankly, I don't think the law of large numbers has a huge role in industry. It is helpful to understand the asymptotic justifications of the common procedures, such as maximum likelihood estimates and tests (including the omniimportant GLMs and logistic regression, in particular), the bootstrap, but these are distributional issues rather than probability of hitting a bad sample issues.
Beyond the topics already mentioned (GLM, inference, bootstrap), the most common statistical model is linear regression, so a thorough understanding of the linear model is a must. You may never run ANOVA in your industry life, but if you don't understand it, you should not be called a statistician.
There are different kinds of industries. In pharma, you cannot make a living without randomized trials and logistic regression. In survey statistics, you cannot make a living without Horvitz-Thompson estimator and non-response adjustments. In computer science related statistics, you cannot make a living without statistical learning and data mining. In public policy think tanks (and, increasingly, education statistics), you cannot make a living without causality and treatment effect estimators (which, increasingly, involve randomized trials). In marketing research, you need to have a mix of economics background with psychometric measurement theory (and you can learn neither of them in a typical statistics department offerings). Industrial statistics operates with its own peculiar six sigma paradigms which are but remotely connected to mainstream statistics; a stronger bond can be found in design of experiments material. Wall Street material would be financial econometrics, all the way up to stochastic calculus. These are VERY disparate skills, and the term "industry" is even more poorly defined than "academia". I don't think anybody can claim to know more than two or three of the above at the same time.
The top skills, however, that would be universally required in "industry" (whatever that may mean for you) would be time management, project management, and communication with less statistically-savvy clients. So if you want to prepare yourself for industry placement, take classes in business school on these topics.
UPDATE: The original post was written in February 2012; these days (March 2014), you probably should call yourself "a data scientist" rather than "a statistician" to find a hot job in industry... and better learn some Hadoop to follow with that self-proclamation.
Alex, I can't comment specifically on Germany or Switzerland, but I do work for an international company with a staff of over 100,000 people from all different countries. Most of these people have at least graduate level degrees, many have Masters and PhDs and, except for the HR and Admin staff most of us are expert in one or more different scientific domains. I have more than 30 years experience, have worked as a skilled scientific / technical specialist, a manager, a Project manager and eventually returned to a purely scientific role that I enjoy. I have also been involved with hiring staff and perhaps some of my observations that follow may be of value to you.
Most new graduates really don't know exactly what they want and it usually takes a few years to find out. In most cases their workplace experience turns out to be quite different compared to what they had expected for a range of reasons. Some workplaces are exciting while some are dull, boring and "workplace politics", bad bosses, etc can sometimes be big problems. A higher degree may or may not help at all with any of these issues.
Most employers want people who can "do the job" and be productive as soon as possible. Higher degrees may or may not matter, depending on the employer. In some situations the door is closed UNLESS you have a PhD. In other situations, the door may be closed BECAUSE you have a PhD and the employer wants someone "less theoretical and with more practical experience".
A PhD does not necessarily mean faster promotions or even much difference in salary and may or may not make any difference to the sort of position that you can obtain. Generally when I have been interviewing candidates, I have been most interested in finding people with relevant work-related experience. A PhD might be a final deciding factor in securing a position, IF the candidate's thesis topic is specifically relevant.
People tend to change jobs more often now than they used to in the past. Your age divided by 2*pi is not a bad rule of thumb for a good number of years to stay in a job before you start going around in circles. Some people work for a while and then return to higher studies. Some people (like me) start on a PhD and then get an "offer too good to refuse" and leave the PhD to go and work. Am I sorry I did that? NO, not at all, and if I were starting over again I would do a PhD in a completely different topic anyway.
The best suggestion that I can give you is to do what you most enjoy doing and see how it unfolds. No-one else can tell you what will be best for you. Sometimes you just have to try something and, if it doesn't work out, then learn as much as you can from it and move on to something else. As Rodin said: Nothing is ever a waste of time if you use the experience wisely.
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The area I am most interested in is the realm of biostatistics. Statistics can be used in this regard to do anything from summarize the results of a drug trial, determining whether Prozac really is more effective than the placebo sugar pill, to tumor detection in cancer patients. Please check out this presentation I found:
What is Biostatistics?
Remember, a statistician is a function that maps a set of data to a set of decisions.