What are the chances of a third world war? Which countries are likely to be involved?
There is really no way to give a numerical estimate of the probability of a third world war. Given the repeated failures of pundits to predict the future, the best we can do is look at potential problem areas and try to understand what would cause potential tensions to increase or decrease.
The first point of military tension which could result in a world war is the South China Sea. China's claim to the "nine-dashed line" would make the South China Sea essentially a Chinese lake. The areas claimed violate both international law and the territorial claims of Vietnam, Japan, and the Philippines. The United States is interested in enforcing freedom of navigation in the area. Although diplomats are working on defusing tensions, it is not impossible that the ongoing tensions between fishing boats and military craft might erupt into a war.
The next obvious point of tension is Russia's increased military opportunism in the areas of the former Soviet Union, especially the Ukraine. This puts Russia in immediate danger of conflict with NATO.
The Middle East is an ongoing area of tension, with Libya, Syria, and Yemen currently war zones. On a positive side, Iran, long considered a rogue state, seems to be willing to give up breakout capacity for developing nuclear weapons in exchange for lifting of sanctions. The real positive is that this means that Iran is now actively engaged in diplomacy with the west, and has a strong economic motive to remain engaged in a positive fashion, rather than acting purely as a destabilizing force. On the other hand, the region is still rent by ethnic and religious strife, and unlikely to achieve peace in the foreseeable future.