Why don't we take a random sample of the population to estimate the amount of COVID-19 cases?

One technique that is also being trialled is testing wastewater/sewerage for RNA from the SARS-CoV2 virus. This won't help you identify who has the virus, but it could inform a city that had previously eliminated the virus that someone has brought the virus back again.

It is being done. The Czech Republic is currently preparing such a study }to begin on the 20th April) to find out the number of asymptomatic patients to better predict the future evolution of the number of infected people. A bit older source in English https://news.expats.cz/coronavirus-in-the-czech-republic/czech-covid-19-central-control-team-plans-blanket-testing-in-sample-of-prague-population/ Much more updated information exists in Czech https://www.seznamzpravy.cz/clanek/prazdniny-podle-prymuly-v-ceskem-hotelu-s-vylety-po-hradech-a-bez-festivalu-99893

"Testování vzorku 17 tisíc lidí v České republice na protilátky proti koronaviru začne v pondělí 20. dubna, do dalšího pondělí budeme znát výsledky." "Testing of the sample of 17 thousand people in the Czech Republic for antigens against coronavirus will begin on Monday 20th April. By the next Monday we will know the results."

Denmark did this recently and found that 2.7% of Danes living in the Capital region already have antibodies for the disease. This isn't randomized per se as these samples are from people donating blood, but so far it's as good as it gets.

why not take a random sample of a few thousand people people, and test all of them to see how many of them have the virus?

Iceland did it, see https://english.alarabiya.net/en/features/2020/03/25/Coronavirus-Iceland-s-mass-testing-finds-half-of-carriers-show-no-symptoms:

As of Sunday night, the country’s health authorities and the biotechnology firm deCode Genetics have tested more than 10,300 people. That might not sound like a large number, compared to the around 350,000 Americans who have been tested for coronavirus according to the COVID Tracking Project, but it is a far higher percentage of tests per capita - a ratio Icelandic authorities have claimed is the highest in the world.

But it is not just the numbers of people being tested that is unusual about Iceland’s approach.

Unlike other countries, where people are only tested if they exhibit symptons of coronavirus or have come into contact with known spreaders, the country is testing thousands of people from the general population who don’t exhibit any symptoms of the virus whatsoever – helping to reveal information about the nature of the pathogen and its symptoms.

This was also done in a city in Italy:

In COVID-19, The University of Padua, Veneto Region and the Red Cross tested the populationof Vò, Italy, 3300 people, to establish the natural history of the virus, the transmission dynamicsand categories of risk. " they found >50 of those who tested positive to be asymptomatic” according to Professor Sergio Romagnani.

The same question was asked on politics.SE: Has there been a random survey of a population for COVID-19?

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References:

• {1} 2020-03-25 - Stanford COVID-19 Evidence Service - Addressing COVID-19 Face Mask Shortages [v1.2] https://archive.org/details/20200325stanfordcovid19evidenceserviceaddressingcovid19facemaskshortagesv1.2

Would that not give us a picture of the proportion of people who have the virus?

Yes, but: at the moment, this is probably not worth while in most countries.

• As Fizz explained, tests are scarce, so we need to use them where the outcome of the test does make a difference in treatment and/or for containing the disease/slowing down the spread.
• Right now, the development of Covid-19 case numbers is quite dynamic in many countries. The percentage we find by testing today will be outdated if not tomorrow then by the end of the week.

• In order to get good estimates of a low prevalence, we need to test large numbers of people and we need tests that have both high sensitivity and specificity. Right now, the tests that are used have not undergone a full validation procedure (in order to have them available asap).

  One consequence of that is that while we can know that they work sufficiently well to be useful for testing high risk patients, e.g. for some test we may know that sensitivity is better than 94% and specificity is better than 97%. If we test a general population with a prevalence of 1 in 1000 using a test with 97% specificity, we'll get around 30 false positives and 1 true positive (with 94% sensitivity). In other words, rather than measuring the prevalence of SARS-CoV-2 infections in the general population, we measured 1 - specificity of the test. Unfortunately, we wouldn't know this.

As a rule of thumb, we can measure prevalences that are >> 1 - specificity of the test. So if we expect a prevalence of 1 : 1000, we'd ask for a test that has specificity at least 99.95 % (with that, we'd observe 1.5 positive in 1000). In order to know that a test has a specificity > 99.95 %, we'd need to validate it with at least 10000 truly negative patients all of which would need to be recognized correctly. Right now, the test may have been validated with 100 negative samples. If you want to read more background about this, I have long answers to How accurate are coronavirus tests? and Why are people with COVID-19 symptoms being denied tests in the US?

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Sentinel Samples

While we do not have such a random sampling scheme from the population right now, we have something that goes into this direction here in Germany: samples from so-called sentinel practices. Sentinel practices are medical practices throughout the country that send patient samples to a central lab where they are analyzed for a number of viruses as part of influenza surveillance. The tested viruses now include also SARS-CoV-2.

These samples are not a random sample of the population you ask for:

• they are taken of patients that show up at the doctor's with acute respiratory disease,
• and right now with the further systematic restriction that these are patients that were not thought to be at a particularly high risk of having Covid-19 (those are sent to the SARS-CoV-2 testing).

In week 12 (Mar 16 - 22), 3 of the 193 sentinel samples that were tested for SARS-CoV-2 were positive. That's a prevalence of 1,6 %, 95 % confidence interval roughly being 0.4 - 4 % (in other words, we have an order of magnitude).

I don't have information on the specificity of the SARS-CoV-2 test used for the sentinel samples, but the published data reports two weeks with 0 positive among 191 + 229 = 420 samples, and if we take those as true negatives, specificity would be better than 99 %.

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I expect that studies will be done once the first wave of Covid-19 is over and once properly validated antibody tests are available. In contrast to the RNA tests now (which test an active infection), antibody tests can say whether someone had such an infection recently (possibly as recent as right now) or a while ago.

The specificity requirements are the same regardless of whether RNA or antibodies are tested.

Italy has 60 million people and around 70 thousand (known) cases. That's barely more than 1 in 1000 prevalence. For most countries it's less than this. So, you'd need a huge sample for a good estimate. And those tests/resources are better deployed where suspected cases are, for the time being.

And sensible or not, I've heard this from a colleague, and it's confirmed by the news that Romania's capital Bucharest plans to randomly test a sample of 10,500 people to determine the extent of the virus spread over there.

Manager of the “Matei Bals” Infectious Diseases Institute Adrian Streinu-Cercel has announced that a pilot project for testing 10,500 persons for the novel coronavirus will start in Bucharest within a scientific study to detect those infected with SARS CoV-2 virus, from the desire to thus prevent the severe forms of illness.

Streinu-Cercel mentioned that, for a population of approximately 2 million inhabitants, the testing of 9,558 persons is necessary, with a correction of 10% being applied to this number. He also stated that the sample can be resized along the way. In this regard, Streinu-Cercel referred to the recommendations of the World Health Organization, which asked that testing be conducted to detect COVID 19. The manager of the “Matei Bals” Institute showed that it is necessary to test the medical staff and patients, but also the population, in order to know if “it’s healthy, if it is currently infected or went through the infection” with this virus, these three pieces of information completely changing how this pandemic is approached in the near future. [...]

Streinu-Cercel underscored that, in lack of screening, the mild forms can go undetected, generating subsequent forms of severe infections. He also mentioned that the Minister of Health approved this project and stressed that this is a study with “scientific value,” not a simple testing. At the same time, he mentioned that in Bucharest there are 188 people diagnosed positively.

Note that ~200 cases in 2M is 1:10,000.

Ha, ha, the more amusing part is that this turned out to be "fake news!" But it's actually informative to read the "retraction/information" as to why they are not actually doing such a study [obviously: insufficient testing capacity for the proposed sample.]

After the manager of the “Matei Bals” Institute for Infectious Diseases, Adrian Streinu-Cercel and Bucharest mayor Gabriela Firea had announced on Thursday that 10,514 people from 5 different age groups in Bucharest will be tested for the novel Coronavirus, PM Ludovic Orban reacted and said this is not possible.

“We cannot establish the people to be tested randomly. Selecting 10,000 are just stories, there is a priority list for testing, it depends on how the situation and the testing capacity are developing,” the PM told a conference at the Health Ministry’s HQs.

Orban said that the medical staff and those targeted by the epidemiological inquiries are qualifying for testing first, adding that “Romania has not been ready and the testing capacity has been limited”.

The premier said that “the testing capacity in Romania will increase up to 2,000 next week” and will gradually rise, but to select 10,000 people “are just stories”, the more the death toll has been low in Romania.

In older news I see that India conducted a random test on 500 of their citizens. Rather predictably, they all came out negative.

And for some more official objections of the same kind DW reported on Apr 7...

Germany's center for disease control, the Robert Koch Institute, has criticized Germany's methods of testing, complaining for example that too many asymptomatic individuals were being tested. The RKI called for an end to this practice on the grounds that Germany could risk running out of tests. Therefore, asymptomatic people are currently not being recommended for testing.

By the way, if what you want to find out are the proportion of asymptomatic cases, there are some studies (albeit on fairly contained populations) that may have an answer to that, e.g. one on the Diamond Princess (the cruise ship quarantined off Japan):

[Overall:] Our estimated asymptomatic proportion is at 17.9% (95%CrI: 15.5–20.2%), which overlaps with a recently derived estimate of 33.3% (95% confidence interval: 8.3–58.3%) from data of Japanese citizens evacuated from Wuhan.

[Caveat:] Considering that most of the passengers were 60 years and older, the nature of the age distribution may lead to underestimation if older individuals tend to experience more symptoms.

From a 2nd study on the Diamond Princess, it's informative to read the testing strategy/order:

Overall 3,063 PCR tests were performed among [the 3,711] passengers and crew members. Testing started among the elderly passengers, descending by age.

An interesting counterpoint to the healthcare providers' perspective is that economists seem to strongly agree on the need for random testing (or better said, per some comments which corrected the question, population-representative testing) in order to calibrate the length of the lockdown from their perspective, of limiting "economic damage" caused by the lockdowns themselves.