Example $4.$

A company has recently developed a new generation of computer processors and has obtained a patent for the invention. The company can either sell the patent or use the processors in the production of computers that have the potential to be very successful. If the patent is sold, the company will receive $1,500$ million ISK. If the patent is not sold, it will cost $600$ million ISK to set up a factory and market the computers in Europe. The difference between the selling price and variable cost for each computer sold in Europe is $60,000$ ISK.

If the company decides to produce computers, its profit will depend on how well the computers are marketed in Europe. There is a $40\%$ chance that the initial response to the marketing will be poor, in which case fixed sales will be $20,000$ units. If the response is good, sales in Europe will be $30,000$ units. If the response is good, the company can also choose to market the computers in America, where there is a $30\%$ chance of an additional $40,000$ units sold in America $($totaling $70,000$ units in Europe and America$)$ and a $70\%$ chance of an additional $10,000$ units sold in America. Marketing in America costs $500$ million ISK, and the difference between the selling price and variable cost for each computer sold in America is $40,000$ ISK.

$($a$)$ Draw a decision tree for the problem.

$($b$)$ Calculate the expected profit for each branch in the tree and use the tree to find the best decision for the company.