The chances that a country will win a medal at the Olympic games could be depend upon how many exceptional athletes live in that country. And the number of exceptional athletes might depend solely upon the sheer number of people living in that country.
The graph above is utilizes the datamaps d3 library to illustrate the total number of Olympic medals won per person living in each winning country. Here dark brown refers to countries that win quite a few medals per person while the white refers to countries that win only a few (grey represents countries with no Olympic victories)*. It looks like Northern Europe produces more Olympic athletes per capita than most other countries.
The history of the Olympics is worth mentioning from an analytic perspective primarily because it illustrates how problems such as these are unexpectedly difficult to analyze. While the tradition of the Olympic games has been fairly consistent for the last 100 years or so, just about everything else has changed. This makes it hard to figure out which type of analysis could encompass the history of these games. Obviously each game is hosted in a different city. This means each event is slightly different. The sports have differed quite a bit too. In the 1900 Olympics athletes competed at tug of war. Medals have been inconsistent. More than 5000 gold medals have been awarded since the inception of the gold medal in 1900. Gold medals are made out of a different type of material at each new Olympic games. Finally, countries themselves have been inconsistent. For instance: Yugoslavia is now multiple countries; East and West Germany are now just Germany; the Soviet Union is now Russia. This makes it hard to illustrate medals won by any of these former countries where geographies have changed.
Back to the analysis. So population seems to indicate Northern Europe. But of course other factors matter. There is a theory that the likelihood for a country to win medals at the Olympic games relates to the number of athletes that country sends to the games. So it is not just the number of people living in a country, but the number of people living in that country pursuing athletics. Unfortunately I neither found resources indicating the number of athletes each country sent to the Olympics nor the number of athletes per country. As a Proxy I will use GNP. GNP represents the relative wealth of each country. The thought is that wealthier countries should have more time to devote to leisure and athletics. This should help increase the number of Olympic athletes.
Now it looks as though Russia, Ethiopia, and Kenya produce an exceptionally high number of Olympic athletes relative to the GNP of their respective countries. But GNP is sort of an unrepresentative number. GNP shows total income of the entire country. This figure is also distorted by population. The citizens of two countries with equal wealth and unequal populations will have quite different standards of living.
The image above shows the number of medals won compared to the per capita GNP for each country. Per capita GNP is a measure of the relative standard of living for each country. This should metric should be doubly useful since it contains information both about the income of the country but also the number of citizens. Considering these attributes, it now appears as though China, Ethiopia, and Russia all have the most Olympic athletes. But, Sochi is not over yet- and thanks to the Ice Hockey scandal [upset] there is still plenty of time for the U.S. to beat Canada and take home Gold.
Data Sources: Olympic Results, World Statistics.
*Scales for these charts are indices that range from [1:9], [White:Dark Brown] where 9 is the highest. This is a discretized, normalized, median-centric, logarithmic index that graphs the ratio of two metrics.