One hears lots of silly things said at MR conferences and one of the silliest and oft-repeated refrains is that you can't do surveys with probability samples any more. There are even those who say that you never could. As often as I get the chance I point out that that's total nonsense. Lots of very serious organizations draw high quality probability samples all the time and get very good results. The prime example here in the US is the Current Population Survey, the government survey used as the basis for calculating the unemployment rate each month. Pretty much everything that comes from Pew is based in probability sample surveys as are many of those political polls that we follow so breathlessly every four years.
The concept of a probability sample is very straightforward. The standard definition is a sample for which all members of the frame population have a known, nonzero chance of selection. Unless you have a complex stratified or multi-stage design it's a pretty simple concept. As long as you have a full list of the population of interest to draw on and everyone has a chance to be selected the resulting sample can be said to represent the population of interest. But there are some serious challenges in current practice.
The first is assembling a frame that includes the entire population you want to study. For example, because of the rise of cell phone only households the landline frame that used to contain the phone numbers of well over 90% of US households no longer does. So it has become standard practice to augment the landline frame with a cell phone frame to ensure full coverage of the population. Clients often can supply customer lists that do a good job of covering their full customer base and we can draw good samples from them as well. Online panels are problematic because they use the panel as the frame and it contains only a very small fraction of the total population.
The second major challenge is declining cooperation. While there are studies that show even surveys with alarmingly low response rates can produce accurate estimates, low response rates make everyone nervous, raise fears of representivity and call results into question. The Current Population Survey gets 90% plus and so we trust the employment rate, but that kind of response is very unusual.
There are other challenges as well but I think it's the deterioration of the landline frame and very low response rates that cause some people to think that probability sampling is no longer possible. Anyone willing to spend the time and the money will get very accurate estimates from a probability sample, better than anything they'll get with an online panel or other convenience samples.
As I have written numerous times on this blog, the lure of online has always been that it's fast and cheap, not that it's better. And depending on how the results are to be used the method can be just fine, fit for purpose. But sometimes the problem requires representivity and when it does probability sampling is still the best way to get it.